Title: Completing the Square
1Completing the Square
2Completing the Square
- This is an x. Show me x2.
x2
x
Show me x2 6x
3Completing the Square
x
x
x
x
x
x
x2
x2
Let's Make a Square
4Completing the Square
How many units are required to complete the
square?
9!
The picture is now x2 6x 9,
which factors (x3)(x3) (x3)2
5Lets Try Another One!
Let's Make a Square
x2
x
x
6Completing the Square
How many units are required to complete the
square?
1!
x2
x2
The picture is now x2 2x 1
which factors (x1)(x1) (x1)2
7Last One with Manipulative
Show me x2 8x
Let's Make a Square
x
x
x
x
x
x
x
x
x2
8Completing the Square
Again, how many units are required to complete
the square?
16
So, the picture is now x2 8x 16
which factors (x4)(x4) (x4)2
9Hard One!
Complete the square for x2 18x ___ How many
units are needed?
There are not enough pieces to do this
problem. Can we do it using paper and pencil?
10What is completing the square used for?
- Completing the square is used for all those not
factorable problems!! - It is used to solve these equations for the
variable.
11Rule for Completing the Square
This is now a PTS!
So, it factors into this!
12Example Find the value of c that makes this a
PTS, then write the expression as the square of
a binomial. x2-3xc
13Example Solve by completing the square.
x26x-80
- x26x-80
- x26x8
- x26x___8___
- x26x989
- (x3)217
-
Dont forget Whatever you add to one side of an
equation, you MUST add to the other side!
14More Examples!
- 5x2-10x300
- x2-2x60
- x2-2x-6
- x2-2x__-6__
- x2-2x1-61
- (x-1)2-5
-
-
- 3x2-12x180
- x2-4x60
- x2-4x-6
- x2-4x__-6__
-
- x2-4x4-64
- (x-2)2-2
15Last Example! Write the quadratic function
yx26x16 in vertex form. What is the vertex
of the functions graph?
- yx26x16
- y-16x26x
- y-16__x26x__
- y-169x26x9
- y-7(x3)2
- y(x3)27
- If the equation, in vertex form, is y(x3)27,
then the vertex must be (-3,7).
16Solving Quadratic Equations by Completing the
Square
17Perfect Square Trinomials
- Examples
- x2 6x 9
- x2 - 10x 25
- x2 12x 36
18Creating a Perfect Square Trinomial
- In the following perfect square trinomial, the
constant term is missing.
X2 14x ____
- Find the constant term by squaring half the
coefficient of the linear term. - (14/2)2
X2 14x 49
19Perfect Square Trinomials
- Create perfect square trinomials.
- x2 20x ___
- x2 - 4x ___
- x2 5x ___
100 4 25/4
20Solving Quadratic Equations by Completing the
Square
- Solve the following equation by completing the
square - Step 1 Move quadratic term, and linear term to
left side of the equation
21Solving Quadratic Equations by Completing the
Square
- Step 2 Find the term that completes the square
on the left side of the equation. Add that term
to both sides.
22Solving Quadratic Equations by Completing the
Square
Step 3 Factor the perfect square trinomial on
the left side of the equation. Simplify the
right side of the equation.
23Solving Quadratic Equations by Completing the
Square
- Step 4 Take the square root of each side
24Solving Quadratic Equations by Completing the
Square
- Step 5 Set up the two possibilities and solve
25Completing the Square-Example 2
- Solve the following equation by completing the
square - Step 1 Move quadratic term, and linear term to
left side of the equation, the constant to the
right side of the equation.
26Solving Quadratic Equations by Completing the
Square
Step 2 Find the term that completes the square
on the left side of the equation. Add that term
to both sides. The quadratic coefficient must be
equal to 1 before you complete the square, so you
must divide all terms by the quadratic
coefficient first.
27Solving Quadratic Equations by Completing the
Square
Step 3 Factor the perfect square trinomial on
the left side of the equation. Simplify the
right side of the equation.
28Solving Quadratic Equations by Completing the
Square
Step 4 Take the square root of each side
29Solving Quadratic Equations by Completing the
Square
Try the following examples. Do your work on your
paper and then check your answers.