QE by Factoring & Quadratic Formula.pptx

Factoring the QE

1. Grade 9 – Mathematics Quarter I SOLVING QUADRATIC EQUATIONS BY FACTORING & QUADRATIC FORMULA

2. OBJECTIVES: solve quadratic equations by factoring in the form 𝑎𝑥 2 + bx = 0; and 𝑎𝑥 2 + bx + c = 0. solve quadratic equations using the quadratic formula.

3. 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 = , then either 𝑎 = or 𝑏 = , or both a and b are 0. Factoring Method Write the equation in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0. Factor the left- hand side of the equation. Set each factor equal to zero using the Principle of Zero Products. Solve each resulting linear equation.

4. Solve. 9 𝑥 2 − 4 = Factor. 3𝑥 + 2 3𝑥 − 2 = Set each factor to 0. 3x + 2 = 3𝑥 = −2 2 𝑥 = − 3 3x − 2 = 3x = 2 2 𝑥 = 3 FACTORING THE DIFFERENCE OF TWO SQUARES TAKE NOTE: (a + b) and (a – b) are sum and difference of two terms respectively. Their product a 2 − b 2 is a difference of two squares

5. Solve. 2𝑥 2 − 5𝑥 − 3 = Factor. Set each factor to 0. x − 3 = x = 3 2x + 1 = 2𝑥 = −1 1 𝑥 = − 2 − 3 1 x 2x = + FACTORING GENERAL TRINOMIAL

6. 5 2 y Solve. 4 y 2 + 20 y = − 25 Transform into 𝒂𝒙 𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 . Factor. 5 2 y 4 y 2 + 20 y + 25 = Set each factor to 0. 2 y − 5 = y = − 5 2 FACTORING PERFECT SQUARE TRINOMIAL − − − =

7. Note : When constant is 0, the quadratic equation will be of the form 𝒂𝒙 𝟐 + 𝒃𝒙 = 𝟎. Solve for 3𝑥 2 + 18𝑥 = Factor. Set each factor to 0. Solve for x. 3𝑥 𝑥 + 6 = 3𝑥 = x + 6 = 𝑥 = x = −6 For equations of this forms, one root will always be equal to zero. 3, 18 = 3 𝑥, 𝑥 2 = 𝑥

8. Solve for 4𝑥 2 − 2𝑥 = Factor. 2𝑥 2𝑥 − 1 = Set each factor to 0. Solve for x. 2𝑥 = 𝑥 = 2x − 1 = 2x = 1 4, 2 = 2 𝑥, 𝑥 2 = 𝑥 1 x = 2 The roots of quadratic equation of the form 𝒂 𝑎𝑥 2 + 𝑏𝑥 = are 𝒙 = 𝟎 and 𝒙 = −𝒃 .

9. 𝑆𝑜𝑙𝑣𝑒. 7𝑥 2 + 18𝑥 = 10𝑥 2 + 12𝑥 7𝑥 2 − 10𝑥 2 + 18𝑥 − 12𝑥 = −3𝑥 2 + 6𝑥 = 𝒂 𝒙 = 𝟎 and 𝒙 = −𝒃 . 𝑎 −𝑏 −6 𝑥 = = = 2 −3 𝒙 = 𝟎 𝒙 = 𝟐

10. Solve the equations without factoring, Instead, use the fact that 𝒂 the roots of 𝑎𝑥 2 + 𝑏𝑥 = are 𝒙 = 𝟎 and 𝒙 = −𝒃 . 𝑥 2 + 3𝑥 = 2 𝑥 2 + 8𝑥 = 9𝑥 2 − 𝑥 = 4𝑥 2 − 10𝑥 = thus, 𝑥 = and 𝑥 = −3. thus, 𝑥 = and 𝑥 = −4. 9 thus, 𝑥 = and 𝑥 = 1 . thus, 𝑥 = and 𝑥 = 5 2 .

11. QUADRATIC FORMULA   Write the quadratic equation in general form Substitute the value of a, b, and c in the quadratic formula Solve the quadratic equation

12. Solve: 7𝑥 2 +18𝑥= 10     a = 7; b = 18; c = 10 7𝑥 2 +18𝑥+ 10 = 0               or 1st 2nd 3rd 4th 5th 6th 7th 8th