Section 8.1

Completing the Square

FactoringFactoring

• Before today the only way we had for solving quadratics was to factor.

x2 - 2x - 15 = 0

(x + 3)(x - 5) = 0

x + 3 = 0 or x - 5 = 0

x = -3 or x = 5

x = {-3, 5}

Zero-factorproperty

FactoringFactoring

x2 = 9

x2 - 9 = 0

(x + 3)(x - 3) = 0

x + 3 = 0 or x - 3 = 0

x = -3 or x = 3

x = {-3, 3}

Zero-factorproperty

Square Root PropertySquare Root Property

• If x and b are complex numbers and if x2 = b, then

bx OR bx

Solve each equation. Write radicals in simplified form.

492 k

49k

7k

Square Root Property

Solve each equation. Write radicals in simplified form.

112 b

11b Square Root Property

Radical will not simplify.

Solve each equation. Write radicals in simplified form.

122 c

12c

32c

Solution Set

Square Root Property

Solve each equation. Write radicals in simplified form.

36)2( 2 m

36)2( m

62 m

62 m22 8m62 m22

4m

}4,8{m

Solve each equation. Write radicals in simplified form.

3)4( 2 p

3)4( p

34 p

Solving Quadratic Equations by Solving Quadratic Equations by Completing the SquareCompleting the Square

x2 - 2x - 15 = 0

(x + 3)(x - 5) = 0

x + 3 = 0

or x - 5 = 0

x = -3 or x = 5

x = {-3, 5}

01522 xx

1522 xx

Now take 1/2 of the coefficient of x.

Square it.Add the result to both sides.

11Factor the left.

Simplify the right.16)1( 2 x161 x41x}5,3{x

Square Root Property

Completing the Square1. Divide by the coefficient

of the squared term.

2. Move all variables to one side and constants to the other.

3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.

4. Factor the left hand side and simplify the right.

5. Root and solve.

2 2 24 0x x 1 1 1 1

2 2 24 0x x 2 2 24x x

2

12 1

2

1 1

1 2

1 25x 1

21 25x

Completing the Square1. Divide by the coefficient

of the squared term.

2. Move all variables to one side and constants to the other.

3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.

4. Factor the left hand side and simplify the right.

5. Root and solve.

21 25x

1 5x

1 5x

{ 4,6}x

Completing the Square1. Make the coefficient of

the squared term =1.

2. Move all variables to one side and constants to the other.

3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.

4. Factor the left hand side and simplify the right.

5. Root and solve.

0253 2 xx3333

03

2

3

52 xx

3

2

3

52 xx1 5 5

2 3 6

2 25 5

6 6

36

49

6

52

x

Completing the Square1. Make the coefficient of the squared term =1.

2. Move all variables to one side and constants to the other.

3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.

4. Factor the left hand side and simplify the right.

5. Root and solve.

25 49

6 36x

36

49

6

5x

6

7

6

5x

26

12

6

7

6

5

x

3

1

6

2

6

7

6

5x

3

1,2x

1. Make the coefficient of the squared term =1.

2. Move all variables to one side and constants to the other.

3. Take half of the coefficient of the x term and square it. Then add to both sides of the equation.

4. Factor the left hand side and simplify the right.

5. Root and solve.

2 5 3 0x x 2 5 3x x

1 5 5

2 1 2

252

25 37

2 4x

252

5 37

2 2x

5 37

2x

Deriving The Quadratic Formula

2 0b c

x xa a

Divide both sides by a

2 22

2 2

b b c bx x

a a a a

2 2

2 2

4

2 4 4

b b acx

a a a

2

2

4

2 4

b b acx

a a

2 4

2

b b acx

a

Complete the square by adding (b/2a)2 to both sides

Factor (left) and find LCD (right)

Combine fractions and take the square root of both sides

Subtract b/2a and simplify

2If 0 (and 0 then:),ax bx c a

Another Way to Solve QuadraticsSquare Root Property

When you introduce the radicalyou must use + and - signs.

Recall that we know thesolution set is

x = {-3, 3}

92 x

92 x3x

92 x

92 x

3x