Math 2
Lesson 3-5: Completing the Square
Name ________________________________
Date ________________________________
Learning Goals:
 I can complete the square to rewrite a quadratic expression (ax2+bx+c) with the form a(x − h)2+k.
 I can identify the vertex and scale factor of a quadratic function when written in vertex form.
Below are three different forms of the same quadratic function
Notes:
Standard Form
Factored Form
2
f ( x)  (2 x  7)( x  5)
f ( x)  2 x  3x  35
Vertex Form
f ( x)  2( x  .75)2  36.125
I. In order to write an equation in vertex form, you have to understand what a perfect square trinomial is.
The next section will help you learn this concept.
1. Expand the following factored expressions. Write your answers in standard form: ax2+bx+c
A. (x + 7)2
B. (y − 6)2
C. (x + 10)2
D. (x – 12)2
 How is each constant in the factored form related to the “b” value in the standard form?
 How is each constant in the factored form related to the “c” value in the standard form?
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2. Apply your observations from #1 to fill in the blanks. Part (A) is done for you.
Perfect Square Trinomial
Factored Form
A. x2 + 8x +16
(x + 4)2
B. x2 − 8x +16
___________
C. ___________________
(x – 3)2
D. x2 +10x + 25
___________
E. x 2  18 x  _____
___________
F. x 2  10 x  _____
___________
G. x 2  7 x  _____
___________
Why do you think these trinomials are called perfect square trinomials?
3. Let’s consider the example from the notes on the front of this page:
f ( x)  2( x  .75)2  36.125
Underline the perfect square trinomial in the equation.
Any equation written in vertex form automatically has a perfect square trinomial in it.
Being able to convert a quadratic function from standard form to vertex form requires using a
technique called completing the square – you “complete” the perfect square trinomial.
Example: Rewrite the equation f ( x)  x  10 x  47 in vertex form.
Goal:
Vertex Form:
y = a(x − h) 2 + k
2
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II. Practice:
1. The following problems represent quadratic equations in standard form. Write the equations in
vertex form: y  a( x  h)2  k .
y  x 2  12 x
Vertex: ________________
Max or Min
Stretch/Compress/Neither
y  x2  9x
Vertex: ________________
Max or Min
Stretch/Compress/Neither
h( x)  x 2  14 x  20
Vertex: ________________
Max or Min
Stretch/Compress/Neither
h( x)  x 2  32 x  12
Vertex: ________________
Max or Min
Stretch/Compress/Neither
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Example: Rewrite the equation f ( x)  2 x  8 x  11 in vertex form.
2
2. Change the following into vertex form:
y  3x 2  12 x  1
Vertex: ________________
Max or Min
Stretch/Compress/Neither
y  5x 2  10 x  3
Vertex: ________________
Max or Min
Stretch/Compress/Neither
Lesson 3-5 Homework
Directions:
 Write the following equations in vertex form.
 Then identify the following:
 Vertex
 If the vertex is a max or min
 If the parabola is stretched, compressed or neither.
1.
y  x2  4x  2
3. y   x 2  5x  5
2. y  x 2  14 x  5
4. y  x 2  10 x
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5.
f  x   2 x2  14 x  21
6.
f  x   3x2 18x
7.
y  2 x 2  8 x
8. y  2 x 2  23x  8
9.
y  .02 x 2  5 x
10. y  .4 x 2  5x  10