Solving Equations by factoring

1. Solving Equations by factoring Algebra I Mrs. Stoltzfus

2. Consider this… If 4a = 0, what do you know? • We know that a = 0, because of the multiplication property of 0.

3. Consider this… What if ab = 0? What do you know? This tells us that a = 0 or b = 0!

4. This is the key to solving quadratic equations! The ZERO PRODUCT property states… For all real numbers a and b, ab = 0 if and only if a = 0 or b = 0

5. Apply the Zero Product Property (x – 3) = 0 or (x + 4) = 0 (x – 3)(x + 4) = 0 +3 +3 - 4 - 4 Solve each equation for the x x = 3 or x = -4 Example A #1 Solution Set: {-4, 3}

6. Apply the Zero Product Property 6b = 0 or(b+5) = 0 or (b+2) = 0 6b(b + 5)(b + 2) = 0 6 6 - 5 - 5 -2 -2 Solve each equation b = 0 or b = -5 or b = -2 Example A #2 Solution Set: {-5,-2, 0}

7. Apply the Zero Product Property 7 ≠ 0 or(3a+4) = 0 or (2a- 5) = 0 7(3a+4)(2a- 5) = 0 -4 -4 +5 +5 3a = -4 2a = 5 Solve each equation 3 3 2 2 Example A #3 Solution Set: {-4/3, 5/2}

8. 21(w - 7)(5w + 5) = 0 Try this one on your own Example A #4 Solution Set: {-1, 7}

9. For some equations, you will need to factor first!

10. Put equation in Standard Form ax + b = 0 Linear Equation ax2 + bx + c = 0 Quadratic Equations ax3 + bx2 + cx + d = 0 Cubic Equations 5z2 – 80 = 0 5z2 = 80 5(z2 – 16) = 0 Factor the Polynomial 5(z+4)(z- 4) = 0 Example Apply the Zero Product Property 5≠0 or (z+4)=0 or (z- 4) = 0 z = -4 or z = 4 {-4, 4}

11. Put equation in Standard Form ax + b = 0 Linear Equation ax2 + bx + c = 0 Quadratic Equations ax3 + bx2 + cx + d = 0 Cubic Equations x2 +8x + 7 = 0 x2 +4x= -7 – 4x Factor the Polynomial (x+7)(x+1) = 0 Example B #5 Apply the Zero Product Property (x+7)=0 or (x + 1) = 0 x = -7 or x = -1 Solve the equations {-7, -1}

12. Put equation in Standard Form ax + b = 0 Linear Equation ax2 + bx + c = 0 Quadratic Equations ax3 + bx2 + cx + d = 0 Cubic Equations v2 – 9v + 20 = 0 v2 +20= 9v Factor the Polynomial (v-4)(v-5) = 0 Example B #6 Apply the Zero Product Property (v-4)=0 or (v-5) = 0 V = 4 or v = 5 Solve the equations {4, 5}

13. Put equation in Standard Form ax + b = 0 Linear Equation ax2 + bx + c = 0 Quadratic Equations ax3 + bx2 + cx + d = 0 Cubic Equations • STOP • Be Careful! This one is tricky. • There is no “9 product property,” which means you cannot set each factor equal to 9. • Foil, combine like terms, and put the equation in standard form! (x+3)(x – 5)= 9 Example B #7

14. Put equation in Standard Form ax + b = 0 Linear Equation ax2 + bx + c = 0 Quadratic Equations ax3 + bx2 + cx + d = 0 Cubic Equations x2 – 2x - 15 = 9 -9 -9 x2 – 2x - 24 = 0 (x+3)(x – 5)= 9 Factor the Polynomial (x-6)(x+4) = 0 Example B #7 Apply the Zero Product Property (x-6)=0 or (x+4) = 0 x = 6 or x = -4 Solve the equations {-4, 6}

15. Put equation in Standard Form ax + b = 0 Linear Equation ax2 + bx + c = 0 Quadratic Equations ax3 + bx2 + cx + d = 0 Cubic Equations w3– 10w2 + 21w = 0 w3 + 21w= 10w2 w(w2– 10w + 21) = 0 Factor the Polynomial w(w-7)(w-3) = 0 Example B #8 Apply the Zero Product Property w = 0 or (w-7)=0 or (w-3) = 0 w = 0 or w = 7or w =3 Solve the equations {0,3, 7}