Solving quadratics with square roots
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This document provides information on solving quadratic equations by finding square roots. It discusses how to solve equations by taking the square root of each side, and that the square root of a positive number has two answers: one positive and one negative. Properties of square roots are presented. Examples are provided on simplifying expressions involving square roots using the product and quotient properties. The document also covers rationalizing the denominator to remove radicals in fractions and solving quadratic equations by taking square roots and isolating the variable. Methods for solving word problems involving falling objects using the quadratic formula are demonstrated.
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Solving quadratics with square roots
- 1. 4.5 Solving Quadratic Equations by Finding Square Roots (p. 250-253)
- 2. How would you solve the equation: x2 = 4 (take the square root of each side!) x =4 2 x = 4 x = 2 or - 2 * 2 * Remember, the square root of a positive # has 2 answers! (one + and one -)
- 4. Properties of Square Roots (a>0 and b>0) 1. Product Property – 2. Quotient Property- ab = a * b a a = b b Example: 40 = 4 *10 = 4 * 10 = 2 10 Example: 3 3 3 = = 4 2 4
- 5. Examples 1. 2. 500 = 100 * 5 = 100 * 5 = 10 5 3 12 * 6 = 3 12 * 6 = 3 72 = 3 36 * 2 = 3 * 6 2 = 18 2 3. 25 9 25 = 9 5 = 3
- 6. Rationalizing the Denominator You CANNOT leave a radical in the denominator of a fraction! (the numerator is OK) Just multiply the top & bottom of the fraction by the radical to “rationalize” the denominator.
- 7. More Examples! 1. 25 3 25 = 3 5 * 3 = 3 * 3 = 5 3 9 = 5 3 3 Can’t have a tent in the basement! 2. 2 11 2 = 11 * 11 22 = * 11 121 22 = 11
- 8. Solving Quadratic Equations 1. Solve. 3 - 5x2 = -9 -3 -3 -5x2 = -12 -5 -5 x2 = 12 5 12 x = 5 2 2. Solve. 3(x-2)2=21 3 3 (x-2)2 = 7 ( x − 2) 2 = ± 7 x−2= ± 7 x = 2± 7 12 12 * 5 60 4 *15 ± 2 15 x= = = = = 5 5 5 5*5 25
- 9. More Examples! 3. Solve. 4x -6=42 +6 +6 4x2=48 4 4 x2 = 12 x 2 = ± 12 2 x = ± 4 * 3 = ±2 3 1 5 4. Solve. ( x − 4) 2 = 6 ( x − 4) 2 = 30 ( x − 4) 2 = ± 30 x − 4 = ± 30 x = 4 ± 30
- 10. Falling Objects! • Use h = -16t2 + h0 Height of the object after it has fallen # of seconds after the object is dropped Object’s initial height
- 11. Example • The tallest building in the USA is in Chicago, Illinois. It is 1450 ft. tall. How long would it take a penny to drop from the top of the building to the ground? h = −16t 2 + h0 0 = −16t + 1450 2 − 1450 = −16t 90.625 = t 2 2 90.625 = t 2 t ≈ 9.52 seconds