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The document discusses formulas for multiplying binomial expressions. It introduces the conjugate product formula, which states that for binomials of the form (A + B) and (A - B), their product is (A + B)(A - B) = A^2 - B^2. Examples are provided to demonstrate expanding expressions using this formula, as well as formulas for squaring binomials of the forms (A + B)^2 and (A - B)^2.

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Multiplication Formulas
The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas
The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The Conjugates Product and the Difference of Squares
The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugates Product and the Difference of Squares
The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 Verification: (A + B)(A – B) = For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 Verification: (A + B)(A – B) = A2 – AB + AB – B2 For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 Verification: (A + B)(A – B) = A2 – AB + AB – B2 = A2 – B2 For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 Verification: (A + B)(A – B) = A2 – AB + AB – B2 = A2 – B2 Conjugate Product Difference of Squares For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 Verification: (A + B)(A – B) = A2 – AB + AB – B2 = A2 – B2 For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares Note: The conjugate of (3x + 2) or (3x – 2) is different from the opposite of (3x + 2) which is (–3x – 2). Conjugate Product Difference of Squares
Multiplication Formulas Here are some examples of squaring: (3x)2 =
Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2,
Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 =
Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2,
Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2
Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) (A + B)(A – B) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 (A + B)(A – B) = A2 – B2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 Check this by multiplying, (A + B)2 = (A + B)(A + B)
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 Check this by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 Check this by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2 When squaring, we obtain two identical copies of AB. For conjugates, we obtain AB and –AB which cancels.
Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 Check this by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2 We say that “(A + B)2 is A2, B2, plus twice A*B”, and “(A – B)2 is A2, B2, minus twice A*B”. When squaring, we obtain two identical copies of AB. For conjugates, we obtain AB and –AB which cancels.
Example B. Expand using the formula. a. (3x + 4)2 Multiplication Formulas
Example B. Expand using the formula. a. (3x + 4)2 (A + B)2 Multiplication Formulas
Example B. Expand using the formula. a. (3x + 4)2 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers.
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1)
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 =
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591
Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. The conjugate formula (A + B)(A – B) = A2 – B2 may be used to multiply two numbers of the forms (A + B) and (A – B) where A2 and B2 can be calculated easily. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591
Multiplication Formulas “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1)
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401 b. (50½) 2 = (50 + ½ )2
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401 b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401 b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2 + 2 (½) (50)
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401 b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2 + 2 (½) (50) = 2,500 + 1/4 + 50
Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401 b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2 + 2 (½) (50) = 2,500 + 1/4 + 50 = 2,550¼ We may take advantage of these formulas tor lengthier multiplication.
Example E. Expand using the formula. (3x – 2y + 4) (3x – 2y – 4) Multiplication Formulas
Example E. Expand using the formula. (3x – 2y + 4) (3x – 2y – 4) = [(3x – 2y) + 4] [(3x – 2y) – 4] Multiplication Formulas group into conjugates
Example E. Expand using the formula. (3x – 2y + 4) (3x – 2y – 4) = [(3x – 2y) + 4] [(3x – 2y) – 4] = (3x – 2y)2 – 42 Multiplication Formulas group into conjugates difference of squares
Example E. Expand using the formula. (3x – 2y + 4) (3x – 2y – 4) = [(3x – 2y) + 4] [(3x – 2y) – 4] = (3x – 2y)2 – 42 = (3x)2 – 2(3x)(2y) + (2y)2 – 42 = 9x2 + 12xy + 4y2 – 16 Multiplication Formulas group into conjugates difference of squares
Exercise. A. Calculate. Use the conjugate formula. Multiplication Formulas 1. 21*19 2. 31*29 3. 41*39 4. 71*69 5. 22*18 6. 32*28 7. 52*48 8. 73*67 B. Calculate. Use the squaring formula. 9. 212 10. 312 11. 392 12. 692 13. 982 14. 30½2 15. 100½2 16. 49½2 18. (x + 5)(x – 5) 19. (x – 7)(x + 7) 20. (2x + 3)(2x – 3) 21. (3x – 5)(3x + 5) C. Expand. 22. (7x + 2)(7x – 2) 23. (–7 + 3x )(–7 – 3x) 24. (–4x + 3)(–4x – 3) 25. (2x – 3y)(2x + 3y) 26. (4x – 5y)(5x + 5y) 27. (1 – 7y)(1 + 7y) 28. (5 – 3x)(5 + 3x) 29. (10 + 9x)(10 – 9x) 30. (x + 5)2 31. (x – 7)2 32. (2x + 3)2 33. (3x + 5y)2 34. (7x – 2y)2 35. (2x – h)2

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4 multiplication formulas x

  • 2. The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas
  • 3. The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The Conjugates Product and the Difference of Squares
  • 4. The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugates Product and the Difference of Squares
  • 5. The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
  • 6. The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
  • 7. The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 Verification: (A + B)(A – B) = For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
  • 8. The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 Verification: (A + B)(A – B) = A2 – AB + AB – B2 For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
  • 9. The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 Verification: (A + B)(A – B) = A2 – AB + AB – B2 = A2 – B2 For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
  • 10. The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 Verification: (A + B)(A – B) = A2 – AB + AB – B2 = A2 – B2 Conjugate Product Difference of Squares For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares
  • 11. The most important product-formulas are (The Squares) (A + B)(A + B) = (A + B) 2 (A – B)(A – B) = (A – B)2 (The Conjugates Product) (A + B)(A – B) Multiplication Formulas The two binomials (A + B) and (A – B) are said to be the conjugate of each other. The Conjugate Product: (A + B)(A – B) = A2 – B2 Verification: (A + B)(A – B) = A2 – AB + AB – B2 = A2 – B2 For example, the conjugate of (3x + 2) is (3x – 2), and the conjugate of (2ab – c) is (2ab + c). The Conjugates Product and the Difference of Squares Note: The conjugate of (3x + 2) or (3x – 2) is different from the opposite of (3x + 2) which is (–3x – 2). Conjugate Product Difference of Squares
  • 12. Multiplication Formulas Here are some examples of squaring: (3x)2 =
  • 13. Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2,
  • 14. Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 =
  • 15. Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2,
  • 16. Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2
  • 17. Multiplication Formulas Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 18. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 19. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) (A + B)(A – B) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 20. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 (A + B)(A – B) = A2 – B2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 21. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 22. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 23. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 24. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.
  • 25. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z4 Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas
  • 26. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas
  • 27. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2
  • 28. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 Check this by multiplying, (A + B)2 = (A + B)(A + B)
  • 29. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 Check this by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2
  • 30. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 Check this by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2 When squaring, we obtain two identical copies of AB. For conjugates, we obtain AB and –AB which cancels.
  • 31. Multiplication Formulas Example A. Expand using the formula. a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4 (A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2 = 4x2y2 – 25z Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4. II. Square Formulas (A + B)2 = A2 + 2AB + B2 (A – B)2 = A2 – 2AB + B2 Check this by multiplying, (A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2 We say that “(A + B)2 is A2, B2, plus twice A*B”, and “(A – B)2 is A2, B2, minus twice A*B”. When squaring, we obtain two identical copies of AB. For conjugates, we obtain AB and –AB which cancels.
  • 32. Example B. Expand using the formula. a. (3x + 4)2 Multiplication Formulas
  • 33. Example B. Expand using the formula. a. (3x + 4)2 (A + B)2 Multiplication Formulas
  • 34. Example B. Expand using the formula. a. (3x + 4)2 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
  • 35. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
  • 36. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
  • 37. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
  • 38. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas
  • 39. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2
  • 40. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2
  • 41. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2
  • 42. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas
  • 43. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers.
  • 44. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49
  • 45. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1)
  • 46. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12
  • 47. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499
  • 48. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48
  • 49. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22
  • 50. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496
  • 51. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 =
  • 52. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32
  • 53. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591
  • 54. Example B. Expand using the formula. a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16 (A + B)2 = A2 + 2AB + B2 Multiplication Formulas b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2 = 9a2 – 30ab + 25b2 III. Some Applications of the Formulas We can use the above formulas to help us multiply numbers. The conjugate formula (A + B)(A – B) = A2 – B2 may be used to multiply two numbers of the forms (A + B) and (A – B) where A2 and B2 can be calculated easily. Example C. Calculate. Use the conjugate formula. a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499 b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496 c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591
  • 55. Multiplication Formulas “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
  • 56. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
  • 57. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
  • 58. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
  • 59. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
  • 60. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
  • 61. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas.
  • 62. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492
  • 63. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2
  • 64. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12
  • 65. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1)
  • 66. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100
  • 67. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401
  • 68. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401 b. (50½) 2 = (50 + ½ )2
  • 69. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401 b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2
  • 70. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401 b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2 + 2 (½) (50)
  • 71. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401 b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2 + 2 (½) (50) = 2,500 + 1/4 + 50
  • 72. Multiplication Formulas Example D. Calculate. Use the squaring formulas. a. 512 = (50 + 1)2 = 502 + 12 + 2(50)(1) = 2,500 + 1 + 100 = 2,601 “(A + B)2 is A2, B2, plus twice A*B”, “(A – B)2 is A2, B2, minus twice A*B”. The Squaring Formulas. b. 492 = (50 – 1)2 = 502 + 12 – 2(50)(1) = 2,500 + 1 – 100 = 2,401 b. (50½) 2 = (50 + ½ )2 = 502 + ½ 2 + 2 (½) (50) = 2,500 + 1/4 + 50 = 2,550¼ We may take advantage of these formulas tor lengthier multiplication.
  • 73. Example E. Expand using the formula. (3x – 2y + 4) (3x – 2y – 4) Multiplication Formulas
  • 74. Example E. Expand using the formula. (3x – 2y + 4) (3x – 2y – 4) = [(3x – 2y) + 4] [(3x – 2y) – 4] Multiplication Formulas group into conjugates
  • 75. Example E. Expand using the formula. (3x – 2y + 4) (3x – 2y – 4) = [(3x – 2y) + 4] [(3x – 2y) – 4] = (3x – 2y)2 – 42 Multiplication Formulas group into conjugates difference of squares
  • 76. Example E. Expand using the formula. (3x – 2y + 4) (3x – 2y – 4) = [(3x – 2y) + 4] [(3x – 2y) – 4] = (3x – 2y)2 – 42 = (3x)2 – 2(3x)(2y) + (2y)2 – 42 = 9x2 + 12xy + 4y2 – 16 Multiplication Formulas group into conjugates difference of squares
  • 77. Exercise. A. Calculate. Use the conjugate formula. Multiplication Formulas 1. 21*19 2. 31*29 3. 41*39 4. 71*69 5. 22*18 6. 32*28 7. 52*48 8. 73*67 B. Calculate. Use the squaring formula. 9. 212 10. 312 11. 392 12. 692 13. 982 14. 30½2 15. 100½2 16. 49½2 18. (x + 5)(x – 5) 19. (x – 7)(x + 7) 20. (2x + 3)(2x – 3) 21. (3x – 5)(3x + 5) C. Expand. 22. (7x + 2)(7x – 2) 23. (–7 + 3x )(–7 – 3x) 24. (–4x + 3)(–4x – 3) 25. (2x – 3y)(2x + 3y) 26. (4x – 5y)(5x + 5y) 27. (1 – 7y)(1 + 7y) 28. (5 – 3x)(5 + 3x) 29. (10 + 9x)(10 – 9x) 30. (x + 5)2 31. (x – 7)2 32. (2x + 3)2 33. (3x + 5y)2 34. (7x – 2y)2 35. (2x – h)2