1 - Completing the Square
MCR3U – Quadratics
Date: ______________________________________________
Completing the Square
Recall that a perfect square is a trinomial that can be expressed as:
#$% + '()
If we expand this, it will look like:
$% ) + 2$'% + ')
For example:
#% + 5()
#2% − 3()
If these were perfect square trinomials, what would the middle term be?
1. % ) + ______ + 36 =
2. % ) + ______ + 49 =
3. % ) + ______ + 121 =
4. 9% ) + ______ + 36 =
If these were perfect square trinomials, what would the last term be?
1. % ) + 12% + ______ =
2. % ) − 18% + ______ =
3. % ) + 5% __________ =
4. 4% ) + 12% __________
"Completing the Square" is a method by which we make a complete square by adding a
constant to a quadratic expression.
Page 1 of 6
1 - Completing the Square
MCR3U – Quadratics
For example: % ) + 10% + 15 is NOT a perfect square. It would be a perfect square if the last
number was 25, not 15.
To complete the square, we will add 25, then take it away so that the value of the
expression is not changed:
Now the first part of the expression is a perfect square. We can rewrite this as:
This last expression is the equivalent of our original equation, but it is in a different form
(called vertex form, more on this later).
Complete the square of the following:
1. % ) + 6% − 7
3
2. % ) + 5% + 4
3. 2% ) + 8% + 12
Page 2 of 6
1 - Completing the Square
MCR3U – Quadratics
Converting Standard Form to Vertex Form
Recall:
Standard form of a quadratic
Factored form of a quadratic
5 = $% ) + '% + 6
5 = $#% − 7(#% − 8(
Vertex form of a quadratic 5 = $#% − ℎ() + : where the vertex is located at: #ℎ, :(
Since we can get so much information about a parabola from the vertex form it would be
useful to be able to convert from factored or standard to vertex form.
Note: The value of 'a' is the same in standard form, vertex and factored form. It is not the
same if the equation is of the form: 5 = $#'% − 7(#6% − 8(
Method 1 – Using zeros to convert from Factored to Vertex forms
Example
Determine the vertex of 5 = 2#% + 4(#% − 6( then rewrite the equation in vertex form.
Page 3 of 6
1 - Completing the Square
MCR3U – Quadratics
Method 2 – Go from Standard to Factored first
Example
Determine the vertex of 5 = 3% ) − 6% − 9
Method 3 – Complete the Square
5 = 2% ) + 4% + 7
Find the equation of a parabola with zeros -4 and 18 and an optimum value of 100.
Page 4 of 6
1 - Completing the Square
MCR3U – Quadratics
Homework
1. Find the value of m that makes the expression a perfect square.
a. % ) + 4% + <
b. % ) − 4% + <
c. % ) − 6% + <
2. Each expression is a perfect square. Assuming that ' > 0, find the value of b.
a. % ) + '% + 49
b. % ) + '% + 36
c. % ) + '% + 169
3. Find the values of h and k that make the following equations true.
a. % ) + 4% = #% − ℎ() + :
b. % ) − 6% = #% − ℎ() + :
c. % ) − 10% = #% − ℎ() + :
4. Complete the square for the following equations.
a. % ) + 10% + 12
b. % ) − 14% + 20
c. % ) − 10% + 5
5. What value of c makes the following expressions a perfect square?
a. % ) + % + 6
b. % ) + 3% + 6
c. % ) + 9% + 6
d. % ) + 0.2% + 6
e. % ) − 15% + 6
f. % ) + '% + 6
6. Express the equation in vertex form by completing the square.
a. 5 = 2% ) + 4%
b. 5 = 3% ) − 9%
c. 5 = −% ) + 6%
d. 5 = −3% ) − 12% + 5
e. 5 = 0.2% ) + 2% + 9
f. 5 = −0.1% ) − 0.6% − 0.4
7. Convert the following to vertex form.
a. 5 = #% − 2(#% + 6(
b. 5 = % #% − 2( − 8
c. 5 = #% + 3(#% − 2(
d. 5 = %#% + 4( + 2
e. 5 = % #−3% + 12( + 5
f. 5 = %#2% − 10( − 3
g. 5 = % ) − 4% − 11
h. 5 = −2% ) + 12% − 11
i.
5 = −% ) − 6% − 13
8. Factor the following. Then find the vertex and rewrite each equation in vertex form.
a. 5 = % ) − 6% + 5
b. 5 = −2% ) + 12% − 16
3
c. 5 = ) % ) + % − 4
9. From 1960 to 1990, the average number of cigarettes, C, smoked in one year by each
Canadian over 18 can be modelled by the equation ? = 4024.5 + 51.48 − 3.18 ) , where t
is the number of years since 1960.
a. When was cigarette consumption the highest?
b. What was the average per capita consumption in that peak year?
Page 5 of 6
1 - Completing the Square
MCR3U – Quadratics
Answers
1.
a. 4
b. 4
c. 9
2.
a. 14
b. 12
c. 26
3.
a. ℎ = −2, : = −4
b. ℎ = 3, : = −9
c. ℎ = 5, : = −25
4.
a. #% + 5() − 13
b. #% − 7() − 29
c. #% − 5() − 20
5.
a. 0.25
b. 2.25
c. 20.25
d. 0.01
e. 56.25
f.
6.
a. 5 = 2#% + 1() − 2
b. 5 = 3#% − 1.5() − 6.75
c. 5 = −#% − 3() + 9
d. 5 = −3#% + 2() + 17
e. 5 = 0.2#% + 5() + 4
f. 5 = −0.1#% + 3() + 0.5
a. 5 = #% + 2() − 16
b. 5 = #% − 1() − 9
c. 5 = #% + 0.5() − 6.25
d. 5 = #% + 2() − 2
e. 5 = −3#% − 2() + 17
f. 5 = 2#% − 2.5() − 15.5
g. 5 = #% − 2() + 15
h. 5 = −2#% − 3() + 7
i.
a. 5 = #% − 1(#% − 5(
b. 5 = −2#% − 2(#% − 4(
c. 5 = ) #% − 2(#% + 4(
5 = #% − 3() − 4
5 = −2#% − 3() + 2
@A
4
7.
5 = −#% + 3() − 4
8.
3
3
5 = ) #% + 1() − 4.5
9.
a. April 1968
b. 4238 cigarettes
10.
a. April 1997
b. $5.09
c. $30.89
Page 6 of 6