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Solving Quadratic Equations – Quadratic Formula

• The following shows how to solve quadratic equations using the Quadratic Formula.

A quadratic equation is any equation that can be put into the form:

The quadratic formula is given by:

2 0ax bx c

2 4

2

b b acx

a

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• Example 1:

Note that the equation is already in the proper form.

Identify a, b, and c.

Solve the equation. 2 12 0x x

2 0ax bx c

12c 1a 1b

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Substitute the values of a, b, and c into the quadratic formula.

1

1

12

a

b

c

2 4

2

b b acx

a

2( 1) ( 1) 4(1)( 12)

2(1)x

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2( 1) ( 1) 4(1)( 12)

2(1)x

Simplify1 1 48

2x

1 49

2x

1 7

2x

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Solution:

1 7

2x

8 6,

2 2x

3, 4x

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• Example 2:

First, write the equation in the form …

Identify a, b, and c.

Solve the equation. 2 1 4x x

2 0ax bx c

2 4 1 0x x

1, 4, 1a b c

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Substitute the values of a, b, and c into the quadratic formula.

2 4

2

b b acx

a

24 (4) 4(1)( 1)

2(1)x

1, 4, 1a b c

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Simplify

24 (4) 4(1)( 1)

2(1)x

4 16 4

2x

4 20

2x

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Solution: 2 5x

4 2 5

2x

4 20

2x

2 2 5

2x

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• Example 3:

First, write the equation in the form …

Identify a, b, and c.

Solve the equation. 23 5 2x x

2 0ax bx c

23 2 5 0x x

3, 2, 5a b c

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Substitute the values of a, b, and c into the quadratic formula.

2 4

2

b b acx

a

22 ( 2) 4(3)(5)

2(3)x

3, 2, 5a b c

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Simplify

22 ( 2) 4(3)(5)

2(3)x

2 4 60

6x

2 56

6x

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Solution:

2 2 14

6

ix

2 56

6x

1 14

3 3x i

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