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Completing the Square
Grade 10Lesson 5-5
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Completing the Square
This is an x. Show me x2.x x2
Show me x2 + 6x
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Completing the Square
x2
x2
x x xx x x
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Completing the Square
The picture is now x2 + 6x + 9,
How many units are required to complete the square?
which factors (x+3)(x+3) = (x+3)2
9!
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Let’s Try Another One!
Show me x2 + 2x
x2 x x
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Completing the Square
x2
How many units are required to complete the square?
x2
The picture is now x2 + 2x + 1
which factors (x+1)(x+1) = (x+1)2
1!
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Last One with Manipulative
Show me x2 + 8x
x2 x xx x x x x x
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Completing the Square
Again, how many units are required to complete the square?
16
So, the picture is now x2 + 8x + 16
which factors (x+4)(x+4) = (x+4)2
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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?
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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.
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Rule for Completing the Square
bxx 2
22
2
b
bxx
2
2
bx
This is now a PTS!So, it factors into
this!
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Example: Find the value of c that makes this a PTS, then write the expression as the square of a binomial. x2-3x+c
b=-3
22
2
3
2
b
c
4
9
2
2
3
x
4
932 xx
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Example: Solve by completing the square. x2+6x-8=0
x2+6x-8=0x2+6x=8x2+6x+___=8+___
x2+6x+9=8+9(x+3)2=17
932
6
22
22
b
17 3 x
173x
Don’t forgetDon’t forget: Whatever you add to one side of an equation, you MUST add to the other side!
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More Examples!5x2-10x+30=0x2-2x+6=0
x2-2x=-6x2-2x+__=-6+__
x2-2x+1=-6+1(x-1)2=-5
3x2-12x+18=0x2-4x+6=0x2-4x=-6x2-4x+__=-6+__
x2-4x+4=-6+4(x-2)2=-2
112
2
22
22
b
51 x
51 ix
422
4
22
22
b
22 x
22 ix
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Last Example! Write the quadratic function y=x2+6x+16 in vertex form. What is the vertex of the function’s graph?
y=x2+6x+16y-16=x2+6xy-16+__=x2+6x+__
y-16+9=x2+6x+9y-7=(x+3)2
y=(x+3)2+7
If the equation, in vertex form, is y=(x+3)2+7, then the vertex must be (-3,7). 93
2
6
22
22
b
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Solving Quadratic Equations by
Completing the Square
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Perfect Square Trinomials
Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36
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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
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Perfect Square Trinomials
Create perfect square trinomials.
x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___
100
4
25/4
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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
2 8 20 0x x
2 8 20x x
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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.
2 8 =20 + x x 21
( ) 4 then square it, 4 162
8
2 8 2016 16x x
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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.
2 8 2016 16x x
2
( 4)( 4) 36
( 4) 36
x x
x
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Solving Quadratic Equations by Completing the Square
Step 4: Take the square root of each side
2( 4) 36x
( 4) 6x
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Solving Quadratic Equations by Completing the Square
Step 5: Set up the two possibilities and solve
4 6
4 6 an
d 4 6
10 and 2 x=
x
x x
x
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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.
22 7 12 0x x
22 7 12x x
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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.
2
2
2
2 7
2
2 2 2
7 12
7
2
=-12 +
6
x x
x x
xx
21 7 7 49
( ) then square it, 2 62 4 4 1
7
2 49 49
16 1
76
2 6x x
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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.
2
2
2
76
2
7 96 49
4 16 16
7 47
4
49 49
16 1
16
6x x
x
x
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Solving Quadratic Equations by Completing the Square
Step 4: Take the square root of each side
27 47( )
4 16x
7 47( )
4 4
7 47
4 4
7 47
4
x
ix
ix
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Solving Quadratic Equations by Completing the Square
2
2
2
2
2
1. 2 63 0
2. 8 84 0
3. 5 24 0
4. 7 13 0
5. 3 5 6 0
x x
x x
x x
x x
x x
Try the following examples. Do your work on your paper and then check your answers.
1. 9,7
2.(6, 14)
3. 3,8
7 34.
2
5 475.
6
i
i