Solving Equations by factoring Algebra I Mrs. Stoltzfus

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

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

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

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

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

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

EXAMPLE A #4 Solution Set: {-1, 7}

For some equations, you will need to factor first!

EXAMPLE 5z 2 = 80 Put equation in Standard Form ax + b = 0 Linear Equation ax 2 + bx + c = 0 Quadratic Equations ax 3 + bx 2 + cx + d = 0 Cubic Equations Factor the Polynomial Apply the Zero Product Property 5z 2 – 80 = 0 5(z 2 – 16) = 0 5(z+4)(z- 4) = 0 5≠0 or (z+4)=0 or (z- 4) = 0 {-4, 4} z = -4 or z = 4

EXAMPLE B #5 x 2 +4x= -7 – 4x Put equation in Standard Form ax + b = 0 Linear Equation ax 2 + bx + c = 0 Quadratic Equations ax 3 + bx 2 + cx + d = 0 Cubic Equations Factor the Polynomial Apply the Zero Product Property x 2 +8x + 7 = 0 (x+7)(x+1) = 0 (x+7)=0 or (x + 1) = 0 {-7, -1} x = -7 or x = -1 Solve the equations

EXAMPLE B #6 v 2 +20= 9v Put equation in Standard Form ax + b = 0 Linear Equation ax 2 + bx + c = 0 Quadratic Equations ax 3 + bx 2 + cx + d = 0 Cubic Equations Factor the Polynomial Apply the Zero Product Property v 2 – 9v + 20 = 0 (v-4)(v-5) = 0 (v-4)=0 or (v-5) = 0 {4, 5} V = 4 or v = 5 Solve the equations

EXAMPLE B #7 (x+3)(x – 5)= 9 Put equation in Standard Form ax + b = 0 Linear Equation ax 2 + bx + c = 0 Quadratic Equations ax 3 + bx 2 + 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!

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

EXAMPLE B #8 w w= 10w 2 Put equation in Standard Form ax + b = 0 Linear Equation ax 2 + bx + c = 0 Quadratic Equations ax 3 + bx 2 + cx + d = 0 Cubic Equations Factor the Polynomial Apply the Zero Product Property w 3 – 10w w = 0 w(w-7)(w-3) = 0 w = 0 or (w-7)=0 or (w-3) = 0 {0,3, 7} w = 0 or w = 7 or w =3 Solve the equations w(w 2 – 10w + 21) = 0