Completing the Square and the Quadratic Formula

Objective: To solve quadratic equations by completing the

square and using the quadratic formula.

Completing the Square

• When completing the square, we follow the same technique each time.

Completing the Square

• When completing the square, we follow the same technique each time.

1. Variables on one side, constants on the other.

Completing the Square

• When completing the square, we follow the same technique each time.

1. Variables on one side, constants on the other.2. Make sure the x2 is a 1x2.

Completing the Square

• When completing the square, we follow the same technique each time.

1. Variables on one side, constants on the other.2. Make sure the x2 is a 1x2.3. Take half of the middle term and square it.

Completing the Square

• When completing the square, we follow the same technique each time.

1. Variables on one side, constants on the other.2. Make sure the x2 is a 1x2.3. Take half of the middle term and square it.4. Add that number to both sides.

Completing the Square

• When completing the square, we follow the same technique each time.

1. Variables on one side, constants on the other.2. Make sure the x2 is a 1x2.3. Take half of the middle term and square it.4. Add that number to both sides.5. Solve

Completing the Square

• Solve: 0822 xx

Completing the Square

• Solve:

• Add eight to both sides.

0822 xx

822 xx

Completing the Square

• Solve:

• Add eight to both sides.• Take half of 2, square it, and add it to both sides.

0822 xx

18122 xx

822 xx

Completing the Square

• Solve:

• Add eight to both sides.• Take half of 2, square it, and add it to both sides.• Solve.

0822 xx

9)1( 2 x

31 x

Completing the Square

• Solve:

• Add eight to both sides.• Take half of 2, square it, and add it to both sides.• Solve.

0822 xx

9)1( 2 x

31 x31x 31x

2x 4x

You Try

• Solve 01162 xx

You Try

• Solve 01162 xx

1162 xx

911962 xx

You Try

• Solve 01162 xx

1162 xx

911962 xx

20)3( 2 x

203 x

203x 523

543

x

x

Completing the Square

• Solve: 0382 2 xx

Completing the Square

• Solve:

• Subtract 3 from both sides

0382 2 xx

382 2 xx

Completing the Square

• Solve:

• Subtract 3 from both sides• Divide each side by 2

0382 2 xx

382 2 xx

2/342 xx

Completing the Square

• Solve:

• Subtract 3 from both sides• Divide each side by 2• Take half of 4, square it, add to both sides

0382 2 xx

382 2 xx

2/342 xx

42/3442 xx

Completing the Square

• Solve:

• Subtract 3 from both sides• Divide each side by 2• Take half of 4, square it, add to both sides• Complete the square

0382 2 xx

382 2 xx

2/342 xx

42/3442 xx

2/5)2( 2 x

Completing the Square

• Solve:

• Subtract 3 from both sides• Divide each side by 2• Take half of 4, square it, add to both sides• Complete the square• Take the square root and solve.

0382 2 xx

382 2 xx

2/342 xx

42/3442 xx

2/5)2( 2 x

252 x

Completing the Square

• Solve:

• Subtract 3 from both sides• Divide each side by 2• Take half of 4, square it, add to both sides• Complete the square• Take the square root and solve.

0382 2 xx

382 2 xx

2/342 xx

42/3442 xx

2/5)2( 2 x

252 x

252x

You Try

• Solve 01082 2 xx

You Try

• Solve 01082 2 xx

1082 2 xx

542 xx

You Try

• Solve 01082 2 xx

1082 2 xx

542 xx

45442 xx

9)2( 2 x

32 x

32x

1,5x

Vertex Form

• After completing the square, the equation is in what we call vertex form since you can find the vertex from the equation.

Vertex Form

• After completing the square, the equation is in what we call vertex form since you can find the vertex from the equation.

9)2( 2 x

9)2( 2 xy

)9,2( vertex

Example 4

Example 4

The Quadratic Formula

• Given a quadratic equation of the form:

• We can solve it using the quadratic formula:

0,02 acbxax

a

acbb

2

42

The Quadratic Formula

• Solve using the quadratic formula. 932 xx

The Quadratic Formula

• Solve using the quadratic formula.• Set the equation equal to zero.

932 xx

0932 xx

The Quadratic Formula

• Solve using the quadratic formula.• Set the equation equal to zero.• a=1, b=3, c=-9

932 xx

0932 xx

The Quadratic Formula

• Solve using the quadratic formula.• Set the equation equal to zero.• a=1, b=3, c=-9• Use the quadratic equation.

932 xx

0932 xx

)1(2)9)(1(4)3(3 2

The Quadratic Formula

• Solve using the quadratic formula.• Set the equation equal to zero.• a=1, b=3, c=-9• Use the quadratic equation.

932 xx

0932 xx

2

3693)1(2

)9)(1(4)3(3 2

The Quadratic Formula

• Solve using the quadratic formula.• Set the equation equal to zero.• a=1, b=3, c=-9• Use the quadratic equation.

932 xx

0932 xx

2

3693)1(2

)9)(1(4)3(3 2

2

4532

533

2

593

You Try

• Solve using the quadratic formula. 732 xx

You Try

• Solve using the quadratic formula.• Set the equation equal to zero.• a=1, b=-3, c=-7• Use the quadratic equation.

732 xx

0732 xx

2

2893)1(2

)7)(1(4)3(3 2

2

373

Axis of Symmetry

• The axis of symmetry for a parabola is the vertical line .

• This is also the x coordinate of the vertex. Put this back into the equation to find the y coordinate of the vertex.

abx 2

Example 4

Example 4

You Try

• Write the equation for the axis of symmetry and the vertex for the following.

1863 2 xxy

You Try

• Write the equation for the axis of symmetry and the vertex for the following.

1863 2 xxy

axis

x

x

1

)3(26

You Try

• Write the equation for the axis of symmetry and the vertex for the following.

1863 2 xxy

axis

x

x

1

)3(26

21

1863

18)1(6)1(3 2

y

y

y

)21,1( Vertex

Homework

• Page 304• 17-45 odd• Page 311• 11-33 odd