10 4 Solving Quadratic Equations in Factored Form

10. 4 Solving Quadratic Equations in Factored Form ØFactored Form – written as the product of two or more factors ØZero Product Property – if the product of two factors is zero, then at least one of the factors must be zero. 2/12/2022 10. 4 Solving Quadratic Equations in Factored Form 1

Ex. 1. Solve the equation. (x-2)(x-3)=0 x-2 = 0 x-3 = 0 +2 +2 +3 +3 x=2 x=3 2/12/2022 Process: 1. Set each factor equal to zero 2. Solve 10. 4 Solving Quadratic Equations in Factored Form 2

Ex. 2. Solve the equation. a. (x+1)(x-3)=0 x+1 = 0 x-3 = 0 -1 -1 +3 +3 x=-1 x=3 c. (x-5)(x+7)=0 x-5 = 0 x+7 = 0 +5 +5 -7 -7 x=5 x=-7 b. x(x-2)=0 x= 0 x-2 = 0 +2 +2 x=2 d. (x-4)(x+1)=0 x-4= 0 x+1 = 0 +4 +4 -1 -1 x=4 x=-1 2/12/2022 10. 4 Solving Quadratic Equations in Factored Form 3

Ex. 3. Solve the equation. a. (x+5)2=0 x+5 = 0 -5 -5 x=-5 c. (x-4)2=0 x-4 = 0 +4 +4 x=4 b. (x+8)2=0 x+8 = 0 -8 -8 x=-8 d. (x+6)2=0 x+6 = 0 -6 -6 x=-6 2/12/2022 10. 4 Solving Quadratic Equations in Factored Form 4

Ex. 4. Solve the equation. a. (2 x+1)(3 x-2)(x-1)=0 2 x+1=0 3 x-2=0 x-1 = 0 -1 -1 +2 +2 +1 +1 2 x=-1 3 x=+2 x=1 2 2 3 3 x=-1/2 x=2/3 b. (3 x-2)(4 x+3)(x+4)=0 3 x-2=0 4 x+3=0 x+4 = 0 +2 +2 -3 -3 -4 -4 3 x=2 4 x=-3 x=-4 3 3 4 4 x=2/3 x=-3/4 2/12/2022 10. 4 Solving Quadratic Equations in Factored Form 5

Ex. 5. Solve the equation. a. (x-4)(x+6)(4 x+3)=0 x-4=0 x+6=0 4 x+3 = 0 +4 +4 -6 -6 -3 -3 x=4 x=-6 4 x=-3 4 4 x=-3/4 b. (x-3)(3 x+2)(x+6)=0 x-3=0 3 x+2=0 x+6 = 0 +3 +3 -2 -2 -6 -6 x=3 3 x=-2 x=-4 3 3 x=-2/3 2/12/2022 10. 4 Solving Quadratic Equations in Factored Form 6

Ex. 6. Solve the equation. a. (2 x+1)(x-8)2=0 2 x+1=0 x-8=0 -1 -1 +8 +8 2 x=-1 x=8 2 2 x=-1/2 b. (x-3)2(3 x-2)=0 x-3=0 3 x-2=0 +3 +3 +2 +2 x=3 3 x=2 3 3 x=2/3 2/12/2022 10. 4 Solving Quadratic Equations in Factored Form 7

Ex. 7. Sketch the graph. y=(x-3)(x+2) x-3=0 x+2=0 +3 +3 -2 -2 x=3 x=-2 x-coord=3+-2 = 1 2 2 y-coord=(1/2 -3)(1/2+2) =(-2 ½)(2 ½) =-25/4=-6 ¼ Vertex: (1/2, -6 ¼) 2/12/2022 Process: 1. Find the x-intercepts (set equal to zero) 2. Find the average – this will be the x-coordinate of the vertex 3. Substitute into the original to find the y-coordinate 4. Sketch the graph 10. 4 Solving Quadratic Equations in Factored Form 8

Ex. 8. Sketch the graph. y=(x-4)(x+2) x-4=0 x+2=0 +4 +4 -2 -2 x=4 x=-2 x-coord=4+-2 = 2=1 2 2 y-coord=(1 -4)(1+2) =(-3)(3) =-9 Vertex: (1, -9) 2/12/2022 10. 4 Solving Quadratic Equations in Factored Form 9

Ex. 9. Sketch the graph. y=x(x+2) x=0 x+2=0 -2 -2 x=-2 x-coord=0+-2 = -2=-1 2 2 y-coord=(-1)(-1+2) =(-1)(1) =-1 Vertex: (-1, -1) 2/12/2022 10. 4 Solving Quadratic Equations in Factored Form 10

Ex. 10. Sketch the graph. y=(x+4)(x-5) x+4=0 x-5=0 -4 -4 +5 +5 x=-4 x=5 x-coord=-4+5 = 1 2 2 y-coord=(1/2+4)(1/2 -5) =(4 1/2)(-4 1/2) =-81=-20 1/4 4 Vertex: (1/2, -20 1/4) 2/12/2022 10. 4 Solving Quadratic Equations in Factored Form 11