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10.4 Solving Quadratic Equations

Objective: The learner will..,NCSCOS

- Solve quadratic equations by factoring, completing the square finding the vertex, AOS, and zeros.
- Find the two ordered pairs for the intersection of the parabola and the linear equation.

- 1.01, 4.02

Factor, find the AOS, and the Vertex:

y = x2– 6x + 8

x2– 6x + 8 = 0

x2– 6x + 8 = 0

(x– 2)(x– 4) = 0

(3)2– 6(3) + 8 = 0

x = 2 x = 4

(x– 2) = 0 (x– 4) = 0

-1

9 – 18 + 8 =

Find the AOS:

vertex: (3, )

-1

2 + 4

Find the vertex form:

=3

x =

2

y = (x – 3)2–1

AOS:3

Factor, find the AOS, and the Vertex:

y = x2 + 2x – 24

x2 + 2x –24 = 0

x2 + 2x – 24 = 0

(x + 6)(x– 4) = 0

(–1)2 + 2(–1) – 24 = 0

x = 4 x = –6

(x– 4) = 0 (x + 6) = 0

–25

1 – 2 – 24 =

Find the AOS:

–25

vertex: (-1, )

4 – 6

Find the vertex form:

=–1

x =

2

y = (x+1)2–25

AOS:–1

Factor, find the AOS, and the Vertex:

y = x2– 6x + 5

x2– 6x + 5 = 0

x2– 6x + 5 = 0

(3)2– 6(3) + 5 = 0

(x– 1)(x– 5) = 0

-4

9 – 18 + 5 =

x = 1 x = 5

(x– 1) = 0 (x– 5) = 0

vertex: (3, )

-4

Find the AOS:

Find the vertex form:

1 + 5

=3

x =

2

y = (x – 3)2–4

AOS:3

10.4 Solving Quadratic Equations

Complete the Square to find the vertex, AOS, and zeros:y = x2 – 4x + 1

y = x2 – 6x + 7

(x2 – 4x+ 4) + 1 – 4 = 0

(x2 – 6x+ 9) + 7 – 9 = 0

(x – 2)2 – 3 = 0

(x – 3)2 – 2 = 0

Vrtx:

(2, –3)

AOS:

Vrtx:

2

(3, –2)

3

AOS:

(x – 2)2 – 3 = 0

(x – 3)2 – 2 = 0

(x – 2)2 = 3

(x – 3)2 = 2

x – 2 = ± 1.73

x – 3 = ± 1.41

x = 3.7 & .27

x = 4.41 & 1.59

10.4 Solving Quadratic Equations

Complete the Square to find the vertex, AOS, and zeros:y = x2 – 8x + 3

y = x2 – 10x + 23

(x2 – 8x + 16) + 3 – 16 = 0

(x2 – 10x + 25) + 23 – 25 = 0

(x – 4)2 – 13 = 0

(x – 5)2 – 2 = 0

Vrtx:

(4, -13)

AOS:

4

(5, -2)

Vrtx:

5

AOS:

(x – 4)2 – 13 = 0

(x – 5)2 – 2 = 0

(x – 4)2 = 13

(x – 5)2 = 2

x – 4 = ± 3.61

(x – 5) = ±1.41

x = 7.6 & .39

x = 6.4 & 3.6

10.4 Solving Quadratic Equations

Complete the Square to find the vertex, AOS, and zeros:x2 – 8x + 3 = 0

x2 + 6x + 4 = 0

(x2 – 8x + 16) + 3 – 16 = 0

(x2 + 6x + 9) + 4 – 9 = 0

(x – 4)2 – 13 = 0

(x + 3)2 – 5 = 0

4

(4, –13)

AOS:

Vrtx:

-3

(-3, -5)

Vrtx:

AOS:

(x – 4)2 – 13 = 0

(x + 3)2 – 5 = 0

(x – 4)2 = 13

(x + 3)2 = 5

x – 4 = 3.61

x + 3 = 2.24

x = .39 & 7.6

x = -0.76 & -5.24

10.4 Solving Quadratic Equations

Complete the Square to find the vertex, AOS, and zeros:x2 + 2x – 7 = 0

x2 – 4x – 1 = 0

(x2 + 2x + 1) – 7 – 1 = 0

(x2 – 4x + 4) – 1 – 4 = 0

(x + 1)2 – 8 = 0

(x – 2)2 – 5 = 0

-1

Vrtx:

(2, -5)

(-1, -8)

2

Vrtx:

AOS =

AOS:

(x – 2)2 – 5 = 0

(x + 1)2 – 8 = 0

(x – 2)2 = 5

(x + 1)2 = 8

x – 2 = 2.24

x + 1 = 2.83

x = -0.24 & 4.24

x = -3.83 & 1.83

10.4 Solving Quadratic Equations

Find the two ordered pairs for the intersection of the parabola and the linear equation.y = x2 + 3x – 15 and y = -5

x2 + 3x – 15 = -5

x2 + 3x – 10 = 0

y = -5

-5

(-5, )

(x + 5)(x – 2) = 0

(x + 5) = 0

x = -5

y = -5

(x – 2) = 0

x = 2

(2, )

-5

10.4 Solving Quadratic Equations

Find the vertex, AOS, and x-intercepts:y = x2 + 3x – 15 and y = -5

(x2 + 3x + 2.25) – 15 – 2.25

(x + 1.5)2 – 17.25

Vrtx: (–1.5, –17.25)

AOS: –1.5

x2 + 3x – 15 = 0

(x + 5)(x – 3) = 0

(2,-5)

(-5,-5)

x = –5 & 3

(2, -5)

(-5, -5)

10.4 Solving Quadratic Equations

Find the two ordered pairs for the intersection of the parabola and the linear equation.y = x2 – 10x + 21 and y = x – 3

y = x – 3

x2 – 10x + 21 = x – 3

y = 8 – 3

x2 – 11x + 24 = 0

y = 5

(x – 8)(x – 3) = 0

5

(8, )

(x – 8) = 0

x = 8

y = x – 3

x = 3

(x – 3) = 0

y = 3 – 3

y = 0

0

(3, )

10.4 Solving Quadratic Equations

Find vertex, AOS, and x-intercepts then graph the parabola and linear equation:y = x2 – 10x + 21 and y = x – 3

(x2 – 10x + 25) + 21 – 25

(x – 5)2 + 21 – 25

(x – 5)2 – 4 = 0

Vrtx: (5, –4)

(8,5)

AOS: 5

(x – 5)2 = 4

(3,0)

x – 5 = 2

x = 7 & 3

(3, 0) (8, 5)

10.4 Solving Quadratic Equations

Find the two ordered pairs for the intersection of the parabola and the linear equation.y = x2 – 4x + 1 and y = -2x + 4

y = -2x + 4

x2 – 4x + 1 = -2x + 4

y = -2(-1) + 4

x2 – 4x + 2x + 1 – 4 = 0

y = 6

x2 – 2x – 3 = 0

(-1, )

6

(x + 1)(x – 3) = 0

y = -2x + 4

(x + 1) = 0

x = -1

y = -2(3) + 4

x = 3

(x – 3) = 0

y = -2

-2

(3, )

10.4 Solving Quadratic Equations

Find vertex, AOS, and x-intercepts then graph the parabola and linear equation:y = x2 – 4x + 1 y = -2x + 4

(x2 – 4x + 4) + 1 – 4

(x – 2)2 + 1 – 4

(-1,6)

(x – 2)2 – 3 = 0

V (2, –3)

(x – 2)2 = 3

(3,-2)

x – 2 = 1.73

x = 3.73 & .27

(-1, 6) (3, -2)

10.4 Solving Quadratic Equations

Find the two ordered pairs for the intersection of the parabola and the linear equation.y = x2 – 4x + 11 and y = x + 5

y = x + 5

x2 – 4x + 11 = x + 5

y = (2) + 5

x2 – 5x + 6 = 0

y = 7

(2, )

7

(x – 2)(x – 3) = 0

y = x + 5

(x – 2) = 0

x = 2

y = (3) + 5

(x – 3) = 0

x = 3

y = 8

(3, )

8

10.4 Solving Quadratic Equations

Find vertex, AOS, and x-intercepts then graph the parabola and linear equation:y = x2 – 4x + 11 and y = x + 5

(x2 – 4x + 4) + 11 – 4

(x – 2)2 + 11 – 4

(x – 2)2 + 7 = 0

Vrtx: (2, 7)

(3,8)

AOS: 2

(2,7)

(x – 2)2 = –7

Can’t take the sq root; no x-intercepts

(2, 7) (3, 8)

10.4 Solving Quadratic Equations

Find the two ordered pairs for the intersection of the parabola and the linear equation.y = x2 – 7x + 10 and y = x – 2

y = x – 2

x2 – 7x + 10 = x – 2

y = 6 – 2

x2 – 8x + 12 = 0

y = 4

(x – 6)(x – 2) = 0

4

(6, )

(x – 6) = 0

x = 6

y = x – 2

x = 2

(x – 2) = 0

y = 2 – 2

y = 0

0

(2, )

10.4 Solving Quadratic Equations

Find vertex, AOS, and x-intercepts then graph the parabola and linear equation:y = x2 – 7x + 10 and y = x – 2

(x2 – 7x + 12.25) + 10 – 12.25

(x – 3.5)2 + 10 – 12.25

(x – 3.5)2 – 2.25 = 0

Vrtx: (3.5, –2.25)

AOS: 3.5

(x – 3.5)2 = 2.25

x – 3.5 = 1.5

x = 5 & 2

(2, 0) (6, 4)

(2,0)

(6,4)

10.4 Solving Quadratic Equations

Find the two ordered pairs for the intersection of the parabola and the linear equation.y = x2 – x – 9 and y = –2x – 3

(–3, 3) and (2, –7)

y = x2 – 4x + 3 and y = x + 3

(0, 3) and (5, 8)

y = x2 – 8x + 5 and y = 2x – 4

(1, –2) and (9, 14)

10.4 Solving Quadratic Equations

Find the two ordered pairs for the intersection of the parabola and the linear equation.y = x2 + 6x + 8 and y = 2x + 8

(–4, 0) and (0, 8)

y = x2 + 4x – 5 and y = −2x – 5

(−6, 7) and (0, −5)

y = x2 – 6x + 9 and y = x – 3

(3, 0) and (4, 1)

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