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Completing the Square

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Completing the Square Grade 10 Lesson 5-5 – PowerPoint PPT presentation

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Title: Completing the Square


1
Completing the Square
  • Grade 10
  • Lesson 5-5

2
Completing the Square
  • This is an x. Show me x2.

x2
x
Show me x2 6x
3
Completing the Square
x
x
x
x
x
x
x2
x2
Let's Make a Square
4
Completing 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
5
Lets Try Another One!
  • Show me x2 2x

Let's Make a Square
x2
x
x
6
Completing 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
7
Last One with Manipulative
Show me x2 8x
Let's Make a Square
x
x
x
x
x
x
x
x
x2
8
Completing 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
9
Hard 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?
10
What 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.

11
Rule for Completing the Square
This is now a PTS!
So, it factors into this!
12
Example Find the value of c that makes this a
PTS, then write the expression as the square of
a binomial. x2-3xc
  • b-3

13
Example 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!
14
More 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

15
Last 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).

16
Solving Quadratic Equations by Completing the
Square
17
Perfect Square Trinomials
  • Examples
  • x2 6x 9
  • x2 - 10x 25
  • x2 12x 36

18
Creating 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

19
Perfect Square Trinomials
  • Create perfect square trinomials.
  • x2 20x ___
  • x2 - 4x ___
  • x2 5x ___

100 4 25/4
20
Solving 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

21
Solving 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.

22
Solving 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.
23
Solving Quadratic Equations by Completing the
Square
  • Step 4 Take the square root of each side

24
Solving Quadratic Equations by Completing the
Square
  • Step 5 Set up the two possibilities and solve

25
Completing 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.

26
Solving 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.
27
Solving 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.
28
Solving Quadratic Equations by Completing the
Square
Step 4 Take the square root of each side
29
Solving Quadratic Equations by Completing the
Square
Try the following examples. Do your work on your
paper and then check your answers.
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