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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.

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objective the learner will

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

10.1 Graphing Parabolas

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 vertex1

10.1 Graphing Parabolas

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 vertex2

10.1 Graphing Parabolas

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

complete the square to find the vertex aos and zeros

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

complete the square to find the vertex aos and zeros1

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

complete the square to find the vertex aos and zeros2

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

complete the square to find the vertex aos and zeros3

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

find the two ordered pairs for the intersection of the parabola and the linear equation

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

find the vertex aos and x intercepts

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)

find the two ordered pairs for the intersection of the parabola and the linear equation1

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, )

find vertex aos and x intercepts then graph the parabola and linear equation

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)

find the two ordered pairs for the intersection of the parabola and the linear equation2

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, )

find vertex aos and x intercepts then graph the parabola and linear equation1

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)

find the two ordered pairs for the intersection of the parabola and the linear equation3

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

find vertex aos and x intercepts then graph the parabola and linear equation2

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)

find the two ordered pairs for the intersection of the parabola and the linear equation4

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, )

find vertex aos and x intercepts then graph the parabola and linear equation3

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)

find the two ordered pairs for the intersection of the parabola and the linear equation5

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)

find the two ordered pairs for the intersection of the parabola and the linear equation6

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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