Factoring Techniques: Difference of Two Squares
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This document contains notes and exercises on factoring the difference of two squares from a Grade 8 mathematics class. It includes examples of factoring various expressions involving differences of two squares. It also provides guidance on determining when an expression can be factored as a difference of two squares and the steps to follow in factoring them. Several practice exercises are provided for students to try factoring differences of two squares on their own.
Factoring Techniques: Difference of Two Squares
- 1. MATHEMATICS 8 Patterns and Algebra (1st Quarter) Factoring the Difference of Two Squares M R . C A R L O J U S T I N O J . L U N A S e c o n d a r y S c h o o l T e a c h e r I
- 2. The Mathematician’s Prayer Heavenly Father, thank You for the blessings You gave unto us, Add joy to the world, Subtract evil from our lives. Multiply the good things for us. Divide the gifts and share them to others. Convert badness to goodness. Help us raise our needs to You. Extract the roots of immoralities and perform our different functions in life. Tell us all that life is as easy as math. Help us all to solve our problems. This we ask in Jesus’ name, the greatest mathematician who ever lived on earth, Amen.
- 3. Add a Footer 3 WARM UP Find the product each expression.
- 4. 4 WARMUP! Find the product of each expression. (𝒂 + 𝟒)(𝒂 − 𝟒) Answer: 𝒂 𝟐 − 𝟏𝟔 MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA
- 5. 5 WARMUP! Find the product of each expression. (𝒎 + 𝟕)(𝒎 − 𝟕) Answer: 𝒎 𝟐 − 𝟒𝟗 MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA
- 6. 6 WARMUP! Find the product of each expression. (𝟐𝒚 + 𝟗)(𝟐𝒚 − 𝟗) Answer: 𝟒𝒚 𝟐 − 𝟖𝟏 MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA
- 7. 7 WARMUP! Find the product of each expression. (𝟑𝒂 + 𝟓𝒃)(𝟑𝒂 − 𝟓𝒃) Answer: 𝟗𝒂 𝟐 − 𝟐𝟓𝒃 𝟐 MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA
- 8. 8 WARMUP! Find the product of each expression. (𝟐𝒎 𝟐 + 𝟑𝒏)(𝟐𝒎 𝟐 − 𝟑𝒏) Answer: 𝟒𝒎 𝟒 − 𝟗𝒏 𝟐 MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA
- 9. MATHEMATICS 8 Patterns and Algebra (1st Quarter) Factoring the Difference of Two Squares M R . C A R L O J U S T I N O J . L U N A S e c o n d a r y S c h o o l T e a c h e r I
- 10. FACTORING CHART MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA This chart will help you determine which method of factoring to use. Type Number of Terms 1. CMF 2 or more 2. Difference of Two Squares 2
- 11. Determine the pattern MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA These are perfect squares! You should be able to list the first 15 perfect squares in 30 seconds… Perfect Squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 1 4 9 16 25 36 … = 12 = 2 = 32 = 42 = 52 = 62 …
- 12. MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA First terms: Outer terms: Inner terms: Last terms: Combine like terms. 𝒙 𝟐– 𝟐𝟓 𝒙 −𝟓 𝒙 +𝟓 𝒙 𝟐 +𝟓𝒙 −𝟓𝒙 −𝟐𝟓 This is called the difference of two squares. 𝒙 𝟐 +𝟓𝒙 −𝟓𝒙 −𝟐𝟓 Review: Multiply (𝒙 − 𝟓)(𝒙 + 𝟓)
- 13. DIFFERENCEOF TWO SQUARES MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA 𝒂 𝟐 − 𝒃 𝟐 = (𝒂 + 𝒃)(𝒂 − 𝒃) or 𝒂 𝟐 − 𝒃 𝟐 = (𝒂 − 𝒃)(𝒂 + 𝒃) The order does not matter!
- 14. DIFFERENCEOF TWO SQUARES MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA Conditions: 1. There are only TWO (2) terms. 2. The numerical coefficient on each term must be a PERFECT SQUARE. 3. The exponents of the literal coefficients are EVEN. 4. The operation separating the two terms is always SUBTRACTION.
- 15. DIFFERENCEOF TWO SQUARES MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA Steps: 1. Get the square root of the first term. 2. Get the square root of the last term. 3. Write the factors as a pair of conjugates. 𝒂 𝟐 − 𝒃 𝟐 = (𝒂 + 𝒃)(𝒂 − 𝒃)
- 16. Example 1 Factor each binomial if possible. 1. 𝒙 𝟐 − 𝟐𝟓 2. 𝒚 𝟐 − 𝟖𝟏 3. 𝒂 𝟐 − 𝟗 𝟐𝟓 4. 𝒃 𝟐 − 𝟖 5. 𝒎 𝟐 + 𝟑𝟔 MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA Factoring Differences of Two Squares = (𝒙 + 𝟓)(𝒙 − 𝟓) = (𝒚 + 𝟗)(𝒚 − 𝟗) = 𝒂 + 𝟑 𝟓 𝒂 − 𝟑 𝟓 prime prime After any common factor is removed, a sum of squares cannot be factored.
- 17. Example 2 Factor each difference of squares. 1. 𝟏𝟔𝒂 𝟐 − 𝟐𝟓 2. 𝟒𝟗𝒃 𝟐 − 𝟏𝟎𝟎 3. 𝟗𝒎 𝟐 − 𝟒𝒏 𝟐 4. 𝟖𝟏𝒙 𝟐 − 𝒚 𝟐 MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA Factoring Differences of Two Squares = 𝟒𝒂 + 𝟓 𝟒𝒂 − 𝟓 = (𝟕𝒃 + 𝟏𝟎)(𝟕𝒃 − 𝟏𝟎) = (𝟑𝒎 + 𝟐𝒏)(𝟑𝒎 − 𝟐𝒏) = (𝟗𝒙 + 𝒚)(𝟗𝒙 − 𝒚) You should always check a factored form by multiplying.
- 18. Example 3 Factor completely. 1. 𝟏𝟖𝒚 𝟐 − 𝟑𝟐 2. 𝟒𝒂𝒃 𝟐 − 𝟗𝒂 MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA Factoring More Complex Differences of Two Squares = 𝟐 𝟗𝒚 𝟐 − 𝟏𝟔 = 𝟐 𝟑𝒚 + 𝟒 (𝟑𝒚 − 𝟒) = 𝒂(𝟒𝒃 𝟐 − 𝟗) = 𝒂 𝟐𝒃 + 𝟑 (𝟐𝒃 − 𝟑) Factor again when any of the factors is a difference of squares as in the last problem. Check by multiplying.
- 19. Example 3 Factor completely. 3. 𝒑 𝟒 − 𝟖𝟏 4. 𝒙 𝟒 − 𝟏𝟔 MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA Factoring More Complex Differences of Two Squares = (𝒑 𝟐 + 𝟗)(𝒑 𝟐 − 𝟗) = (𝒑 𝟐 + 𝟗)(𝒑 + 𝟑)(𝒑 − 𝟑) = (𝒙 𝟐 + 𝟒)(𝒙 𝟐 − 𝟒) = (𝒙 𝟐 + 𝟒)(𝒙 + 𝟐)(𝒙 − 𝟐) Factor again when any of the factors is a difference of squares as in the last problem. Check by multiplying.
- 20. Add a Footer 20 YOU TRY! Factor each expression completely.
- 21. Exercises Factor completely. 1. 𝒂 𝟐 − 𝟏𝟎𝟎 2. 𝒎 𝟐 − 𝟐𝟓 3. 𝒚 𝟐 − 𝟏𝟔 4. 𝒃 𝟐 − 𝟏𝟐𝟏 5. 𝒙 𝟐 − 𝟒𝟗 MATHEMATICS 8 (PATTERNS AND ALGEBRA 1st Quarter) Mr. CARLO JUSTINO J. LUNA Factoring Differences of Two Squares 6. 𝟒𝒂 𝟐 − 𝟖𝟏 7. 𝟗𝒎 𝟐 − 𝟏 8. 𝟐𝟓𝒚 𝟐 − 𝟒𝟗𝒛 𝟐 9. 𝟖𝒃 𝟐 − 𝟓𝟎 10. 𝟏𝟔𝒙 𝟒 − 𝟖𝟏
- 22. MATHEMATICS 8 Patterns and Algebra (1st Quarter) THANK YOU! M R . C A R L O J U S T I N O J . L U N A S e c o n d a r y S c h o o l T e a c h e r I