Solving Quadratic Equations by Factoring
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This document provides examples for solving quadratic equations by factoring. It explains how to solve equations of the form ax^2 + bx = 0 and ax^2 + bx + c = 0 by factoring and setting each factor equal to zero. Some example problems are worked out step-by-step, including solving 11x^2 - 13x = 8x - 3x^2 and 7x^2 + 18x = 10x^2 + 12x. The document also discusses using the fact that the roots of ax^2 + bx = 0 are x = 0 and x = -b/a to solve equations without factoring. It concludes by explaining how to use the zero product property to solve a quadratic
Solving Quadratic Equations by Factoring
- 1. Grade 9 – Mathematics Quarter I SOLVING QUADRATIC EQUATIONS BY FACTORING
- 2. OBJECTIVES: •solve quadratic equations by factoring in the form 𝑎𝑥2 + bx = 0; and •solve quadratic equations by factoring in the form 𝑎𝑥2 + bx + c = 0.
- 3. Note: When constant is 0, the quadratic equation will be of the form 𝒂𝒙 𝟐 + 𝒃𝒙 = 𝟎. Solve for 3𝑥2 + 18𝑥 = 0 Factor. 3𝑥 𝑥 + 6 = 0 Set each factor to 0. 3𝑥 = 0 x + 6 = 0 Solve for x. 𝑥 = 0 x = −6 For equations of this forms, one root will always be equal to zero. 3, 18 = 3 𝑥, 𝑥2 = 𝑥
- 4. Solve for 4𝑥2 − 2𝑥 = 0 Factor. 2𝑥 2𝑥 − 1 = 0 Set each factor to 0. 2𝑥 = 0 2x − 1 = 0 Solve for x. 𝑥 = 0 2x = 1 4, 2 = 2 𝑥, 𝑥2 = 𝑥 x = 1 2 The roots of quadratic equation of the form 𝑎𝑥2 + 𝑏𝑥 = 0 are 𝒙 = 𝟎 and 𝒙 = −𝒃 𝒂 .
- 5. 𝑆𝑜𝑙𝑣𝑒. 11𝑥2 − 13𝑥 = 8𝑥 − 3𝑥2 11𝑥2 + 3𝑥2 − 13𝑥 − 8𝑥 = 0 14𝑥2 − 21𝑥 = 0 𝒙 = 𝟎 and 𝒙 = −𝒃 𝒂 . 𝑥 = −𝑏 𝑎 = −(−21) 14 = 21 14 = 3 2 𝒙 = 𝟎 𝒙 = 𝟑 𝟐
- 6. 𝑆𝑜𝑙𝑣𝑒. 7𝑥2 + 18𝑥 = 10𝑥2 + 12𝑥 7𝑥2 − 10𝑥2 + 18𝑥 − 12𝑥 = 0 −3𝑥2 + 6𝑥 = 0 𝒙 = 𝟎 and 𝒙 = −𝒃 𝒂 . 𝑥 = −𝑏 𝑎 = −6 −3 = 2 𝒙 = 𝟎 𝒙 = 𝟐
- 7. Solve the equations without factoring, Instead, use the fact that the roots of 𝑎𝑥2 + 𝑏𝑥 = 0 are 𝒙 = 𝟎 and 𝒙 = −𝒃 𝒂 . b. 2𝑥2 + 8𝑥 = 0 a. 𝑥2 + 3𝑥 = 0 c. 9𝑥2 − 𝑥 = 0 d. 4𝑥2 − 10𝑥 = 0 thus, 𝑥 = 0 and 𝑥 = −3. thus, 𝑥 = 0 and 𝑥 = −4. thus, 𝑥 = 0 and 𝑥 = 1 9 . thus, 𝑥 = 0 and 𝑥 = 5 2 .
- 8. Factoring may also be used to solve a quadratic equation when none of the constants 𝑎, 𝑏, 𝑜𝑟 𝑐 is 0. ZERO PRODUCT PROPERTY If 𝑎𝑏 = 0, then either 𝑎 = 0 or 𝑏 = 0, or both a and b are 0. Factoring Method 1. Write the equation in the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0. 2. Factor the left-hand side of the equation. 3. Set each factor equal to zero using the Principle of Zero Products. 4. Solve each resulting linear equation.
- 9. Solve. 𝑥2 + 9𝑥 = −8 Transform into 𝒂𝒙 𝟐 + 𝒃𝒙 + 𝒄 = 𝟎. 𝑥2 + 9𝑥 + 8 = 0 Factor. 𝑥 + 1 𝑥 + 8 = 0 Set each factor to 0. x + 1 = 0 x + 8 = 0 𝑥 = −1 x = −8
- 10. Solve.2𝑥2 − 5𝑥 − 3 = 0 Factor. 2𝑥 + 1 𝑥 − 3 = 0 Set each factor to 0. 2x + 1 = 0 x − 3 = 0 2𝑥 = −1 x = 3 𝑥 = − 1 2
- 11. Solve. 9𝑥2 − 4 = 0 Factor. 3𝑥 + 2 3𝑥 − 2 = 0 Set each factor to 0. 3x + 2 = 0 3x − 2 = 0 3𝑥 = −2 3x = 2 𝑥 = − 2 3 𝑥 = 2 3