Post on 26-Mar-2015
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