completing the square ccss: a.sse.3; f.if.7
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Completing the Square CCSS: A.SSE.3; F.IF.7. Solving Quadratics By Completing the Square. Must be a perfect Square. CCSS: A.SSE.3. CHOOSE and PRODUCE an equivalent form of an expression to REVEAL and EXPLAIN properties of the quantity represented by the expression. - PowerPoint PPT PresentationTRANSCRIPT
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Completing the Square
CCSS: A.SSE.3; F.IF.7
Solving Quadratics By
Completing the SquareMust be a perfectSquare
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CCSS: A.SSE.3
CHOOSE and PRODUCE an equivalent form of an expression to REVEAL and EXPLAIN properties of the quantity represented by the expression.
a. FACTOR a quadratic expression to reveal the zeros of the function it defines.
b. COMPLETE THE SQUARE iin a quadratic expression to REVEAL the maximum or minimum value of the function it defines.
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CCSS: F.IF.7
GRAPH functions expressed symbolically and SHOW key features of the graph, by hand in simple cases and using technology for more complicated cases.*
a. GRAPH linear and quadratic functions and show intercepts, maxima, and minima.
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Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the
reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
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Essential Questions
How do I determine the domain, range, maximum, minimum, roots, and y-intercept of a quadratic function from its graph?
How do I use quadratic functions to model data? How do I solve a quadratic equation with non - real
roots?
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2( 5) 64x 5 8x
When you take the square root, You MUST consider the Positive and Negative answers.
5 8x 5 8x 5 5
13x 5 5
3x
PerfectSquare
On One side
Take Square Root
ofBOTH SIDES
2 ( 5) 64x
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PerfectSquare
On One side
Take Square Root
ofBOTH SIDES
But what happens if you DON’T have a perfect square on one side…….
You make it a Perfect Square
Use the relations on next slide…
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2( 6)x ( 2 ) To expand a perfect square binomial:
2 12 36x x 6x 26
We can use these relations to find the missing term….To make it a perfect square trinomial that can be factored into a perfect square binomial.
2 _ _12 _x x 12 2 6 626 36
36
2x
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Take ½ middle term
Then square it
The resulting trinomial is called a perfect square trinomial,
which can be factored into a perfect square binomial.
2 _ _18 _ _x x
18 2 92(9) 81
81 2( 9)x
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1. 2 12 0x x
1. Make one side a perfect square
2. Add a blank to both sides
3. Divide “b” by 2
4. Square that answer.
5. Add it to both sides
6. Factor 1st side
7. Square root both sides
8. Solve for x
2 0x x ___ ___12 2 6
2(6) 36
36 362( 6)x 362( 6) 36x 6 6x
6 6x 6 6x 6 6
12x 6 6
0x
12
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Factor this Perfect square trinomial
2 12 36x x
What is the
Square root
of x2
2( )x
Bring dow
n sign
6W
hat is the S
quare root of 36
2( 6)x
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2. 2 8 0x x
1. Move constant to other side.
2. Add a blank to both sides
3. Divide “b” by 2
4. Square that answer.
5. Add it to both sides
6. Factor 1st side
7. Square root both sides
8. Solve for x
2 8x x ___ ___6 2 3
2(3) 9
9 92( 3)x 12( 3) 1x
3 1x 3 1x 3 1x 3 3
4x 3 3
2x
6
6
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Factor this Perfect square trinomial
2 6 9x x
What is the
Square root
of x 2
2( )x
Bring dow
n sign
3W
hat is the S
quare root of 9
2( 3)x
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3. 2 8 84 0x x
1. Move constant to other side.
2. Add a blank to both sides
3. Divide “b” by 2
4. Square that answer.
5. Add it to both sides
6. Factor 1st side
7. Square root both sides
8. Solve for x
2 84x x ___ ___8 2 4
2(4) 1616 16
2( 4)x 1002( 4) 100x
4 10x 4 10x 4 10x 4 4
14x 4 4
6x
8
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Factor this Perfect square trinomial
2 8 16x x
What is the
Square root
of x 2
2( )x
Bring dow
n sign
4W
hat is the S
quare root of 9
2( 4)x
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4. 2 2 15 0x x
1. Move constant to other side.
2. Add a blank to both sides
3. Divide “b” by 2
4. Square that answer.
5. Add it to both sides
6. Factor 1st side
7. Square root both sides
8. Solve for x
2 15x x ___ ___2 2 1
2(1) 11 1
2( 1)x 162( 1) 16x
1 4x 1 4x 1 4x 1 1
3x 1 1
5x
2
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Factor this Perfect square trinomial
2 2 1x x
What is the
Square root
of x 2
2( )x
Bring dow
n sign
1W
hat is the S
quare root of 9
2( 1)x
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Steps to solve Quadratics by completing the square:
Move the constant to side by itself. Make the side (w/variables) a perfect square by
adding a certain number to both sides. To calculate this number
– Divide “b” (middle term) by 2– Then square that answer
Take the square root of both sides of eq Then solve for x
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In a perfect square, there is a relationship between the coefficient of the middle term and the constant term.
2( 7)x
7 1
(14)2
27 49
2 14 49x x
7
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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
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