1. Completing the Square
Topic 7.2.1

2. Completing the Square 7.2.1 Topic California Standard:
14.0 Students solve a quadratic equation by factoring or completing the square.
What it means for you:
You’ll form perfect square trinomials by adding numbers to binomial expressions.
Key words:
completing the square
perfect square trinomial
binomial

3. Completing the Square 7.2.1 Topic
“Completing the square” is another method for solving quadratic equations — but before you solve any equations, you need to know how completing the square actually works.

4. Completing the Square 7.2.1 Topic
Writing Perfect Square Trinomials as Perfect Squares
An expression such as (x + 1)2 is called a perfect square — because it’s (something)2.
In a similar way, an expression such as x2 + 2x + 1 is called a perfect square trinomial (“trinomial” because it has 3 terms).
This is because it can be written as a perfect square: x2 + 2x + 1 = (x + 1)2
Any trinomial of the form x2 + 2dx + d2 is a perfect square trinomial, since it can be written as the square of a binomial: x2 + 2dx + d2 = (x + d)2

5. Completing the Square 7.2.1 Topic
Converting Binomials to Perfect Squares
The binomial expression x2 + 4x is not a perfect square — it can’t be written as the square of a binomial.
However, it can be turned into a perfect square trinomial if you add a constant (a number) to the expression.

6. Completing the Square 7.2.1 Topic x2 + 2dx + d2 = (x + d)2
Example 1
Convert x2 + 4x to a perfect square trinomial.
Solution
To do this you have to add a number to the original expression.
First look at the form of perfect square trinomials, and compare the coefficient of x with the constant term (the number not followed by x or x2):
x2 + 2dx + d2 = (x + d)2
The coefficient of x is 2d, while the constant term is d2.
The constant term is the square of half of the coefficient of x.
To convert x2 + 4x to a perfect square trinomial, add the square of half of 4 — that is, add 22 = 4, to give x2 + 4x + 4 = (x + 2)2
Solution follows…

7. Completing the Square 7.2.1 Topic Guided Practice
By adding a constant, convert each of these binomials into a perfect square trinomial.
1. x2 + 14x
2. x2 – 12x 3. x2 + 2x
4. x2 – 8x 5. y2 + 20y
6. p2 – 16p
x2 + 14x + 49
x2 – 12x + 36
x2 + 2x + 1
x2 – 8x + 16
y2 + 20y + 100
p2 – 16p + 64
Solution follows…

8. Completing the Square 7.2.1 Topic Completing the Square for x2 + bx
To convert x2 + bx into a perfect square trinomial, add b 2
The resulting trinomial is x b 2

9. Completing the Square 7.2.1 Topic
Example 2
Form a perfect square trinomial from x2 + 8x.
Solution
Here, b = 8, so to complete the square you add 2 8
This gives you x2 + 8x + 16.
Solution follows…

10. Completing the Square 7.2.1 Topic
Example 3
What must be added to y2 – 12y to make it a perfect square trinomial?
Solution
This time, b = –12
To complete the square you add = (–6)2 = 36. 2
–12
So 36 must be added (giving y2 – 12y + 36).
Solution follows…

11. Completing the Square 7.2.1 Topic
Example 4
Suppose x2 – 10x + c is a perfect square trinomial, and is equal to (x + k)2. What are the values of c and k?
Solution
Here the coefficient of x is –10.
So to form a perfect square trinomial, the constant term has to be the square of half of –10, so c = (–5)2 = 25.
Therefore x2 – 10x + c = x2 – 10x + 25
= (x + k)2.
Now multiply out the parentheses of (x + k)2.
(x + k)2 = x2 + 2kx + k2
Solution continues…
Solution follows…

12. Completing the Square 7.2.1 Topic
Example 4
Suppose x2 – 10x + c is a perfect square trinomial, and is equal to (x + k)2. What are the values of c and k?
Solution (continued)
(x + k)2 has to equal x2 – 10x + 25, which gives x2 – 10x + 25 = x2 + 2kx + k2.
Equate the coefficients of x: the coefficient of x on the left-hand side is –10, while on the right-hand side it is 2k.
So –10 = 2k, or k = –5.
Comparing the constant terms in a similar way, you find that 25 = k2 , which is also satisfied by k = –5.

13. Completing the Square 7.2.1 Topic Guided Practice
Find the value of k that will make each expression below a perfect square trinomial.
7. x2 – 7x + k
8. q2 + 5q + k
9. x2 + 6mx + k
10. d2 – 2md + k
k = = 49 4 2 –7
k = = 25 4 2 5
k = = 9m2 2 6m
k = = m2 2 2m
Solution follows…

14. Completing the Square 7.2.1 Topic Guided Practice
Form a perfect square trinomial from the following expressions by adding a suitable term.
11. x2 + 10x
12. x2 – 16x
13. y2 + 2y
14. x2 + bx
15. y2 – 18y
16. a2 – 2a
17. y2 + 12y
18. y2 + 36y
Add a constant equal to (10 ÷ 2)2 Þ x2 + 10x + 25
Add a constant equal to (–16 ÷ 2)2 Þ x2 – 16x + 64
Add a constant equal to (2 ÷ 2)2 Þ y2 + 2y + 1
Add a constant equal to (b ÷ 2)2 Þ x2 + bx + b2 4
Add a constant equal to (–18 ÷ 2)2 Þ y2 – 18y + 81
Add a constant equal to (–2 ÷ 2)2 Þ a2 – 2a + 1
Add a constant equal to (12 ÷ 2)2 Þ y2 + 12y + 36
Add a constant equal to (36 ÷ 2)2 Þ y2 + 36y + 324
Solution follows…

15. Completing the Square 7.2.1 Topic Guided Practice
Form a perfect square trinomial from the following expressions by adding a suitable term.
19. 6x + 9
20. 1 – 8x
– 20y
Add a squared term, so that ax2 + 6x + 9 = (bx + 3)2 Find a and b, ax2 + 6x + 9 = b2x2 + 6bx + 9 Comparing x terms shows b = 1 Þ a = b2 = 1 So the trinomial is x2 + 6x + 9
Add a squared term, so that ax2 – 8x + 1 = (bx – 1)2 Find a and b, ax2 – 8x + 1 = b2x2 – 2bx + 1 Comparing x terms shows b = 4 Þ a = b2 = 16 So the trinomial is 16x2 – 8x + 1
Add a squared term, so that ay2 – 20y + 25 = (by – 5)2 Find a and b, ay2 – 20y + 25 = b2y2 – 10by + 25 Comparing y terms shows b = 2 Þ a = b2 = 4 So the trinomial is 4y2 – 20y + 25
Solution follows…

16. Completing the Square 7.2.1 Topic Guided Practice
Form a perfect square trinomial from the following expressions by adding a suitable term.
22. 4y2 + 4yb
23. 4a2 + 12ab
24. 9a2 + 16b2
Add a b2 term, so that 4y2 + 4yb + a2b2 = (2y + ab)2 Find a, 4y2 + 4yb + a2b2 = 4y2 + 4aby + a2b2 Comparing y terms shows a = 1 Þ a2 = 1 So the trinomial is 4y2 + 4yb + b2
Add a b2 term, so that 4a2 + 12ab + c2b2 = (2a + cb)2 Find c, 4a2 + 12ab + c2b2 = 4a2 + 4cab + c2b2 Comparing a terms shows c = 3 Þ b2 = 9 So the trinomial is 4a2 + 12ab + 9b2
Add an ab term, so that 9a2 + cab + 16b2 = (3a + 4b)2 Find c, 9a2 + cab + 16b2 = 9a2 + 24ab + 16b2 Comparing ab terms shows c = 24 So the trinomial is 9a2 + 24ab + 16b2
Solution follows…

17. Completing the Square 7.2.1 Topic Guided Practice
The quadratics below are perfect square trinomials. Find the value of c and k to make each statement true.
25. x2 – 6x + c = (x + k)2
26. x2 + 16x + c = (x + k)2
27. 4x2 + 12x + c = (2x + k)2
28. 9x2 + 30x + c = (3x + k)2
29. 4a2 – 4ab + cb2 = (2a + kb)2
30. 9a2 – 12ab + cb2 = (3a + kb)2
c = 9, k = –3
c = 64, k = 8
c = 9, k = 3
c = 25, k = 5
c = 1, k = –1
c = 4, k = –2
Solution follows…

18. Completing the Square 7.2.1 Topic
If the Coefficient of x2 isn’t 1, Add a Number
With an expression of the form ax2 + bx, you can add a number to make an expression of the form a(x + k)2.

19. Completing the Square 7.2.1 Topic
Example 5
If 3x2 – 12x + m is equal to 3(x + d)2, what is m?
Solution
Multiply out the parentheses of 3(x + d)2 to get: 3(x + d)2 = 3x2 + 6dx + 3d2
So 3x2 – 12x + m = 3x2 + 6dx + 3d2
Equate the coefficients of x, and the constants, to get: –12 = 6d and m = 3d2
The first equation tells you that d = –2.
And the second tells you that m = 3(–2)2, or m = 12.
So 3x2 – 12x + 12 = 3(x – 2)2.
Solution follows…

20. Completing the Square 7.2.1 Topic Completing the square for ax2 + bx
The expression ax2 + bx can be changed to a trinomial of the form a(x + k)2. b 2
To do this, add = . 1 a b2 4a
The resulting trinomial is a b 2a 2
x +

21. Completing the Square 7.2.1 Topic
Example 6
Convert 2x2 + 10x to a perfect square trinomial.
Solution 2
Here, a = 2 and b = 10, so you add = = 10 1
100 4 25
This gives you 2x2 + 10x 25 2
Solution follows…

22. Completing the Square 7.2.1 Topic Guided Practice
The quadratics below are of the form a(x + d)2. Find the value of m and d in each equation.
31. 5x2 + 10x + m = 5(x + d)2
32. 4x2 – 24x + m = 4(x + d) x2 – 28x + m = 2(x + d)2
34. 3x2 – 30x + m = 3(x + d) x2 + 32x + m = 4(x + d)2
m = 5, d = 1
m = 36, d = –3
m = 98, d = –7
m = 75, d = –5
m = 64, d = 4
Solution follows…

23. Completing the Square 7.2.1 Topic Guided Practice
The quadratics below are of the form a(x + d)2. Find the value of m and d in each equation.
36. 20x2 + 60x + m = 5(2x + d) x2 – 20x + m = 5(2x + d)2
38. 27x2 + 18x + m = 3(3x + d) x2 + 36x + m = 3(3x + d)2
40. 16x2 – 80x + m = 4(2x + d)2
m = 45, d = 3
m = 5, d = –1
m = 3, d = 1
m = 12, d = 2
m = 100, d = –5
Solution follows…

24. Completing the Square 7.2.1 Topic Guided Practice
Add a term to convert each of the following into an expression of the form a(x + k)2.
41. 2x2 – 12x
42. 3a2 + 12a 43. 6y2 – 60y
44. 4x2 – 48x 45. 5x
46. 8x x + 12
x + 100
2x2 – 12x + 18 = 2(x – 3)2
3a2 + 12a + 12 = 3(a + 2)2
6y2 – 60y =6(y – 5)2
4x2 – 48x = 4(x – 6)2
5x2 + 70x = 5(x + 7)2
8x2 + 8x + 2 = 8(x + )2 1 2
27x2 + 63x + 12 = 27(x + )2 2 3
36x x = 36(x + )2 5 3
Solution follows…

25. Completing the Square 7.2.1 Topic Independent Practice
Find the value of c that will make each expression below a perfect square trinomial.
1. x2 + 9x + c
2. x2 – 11x + c 3. x2 + 12xy + c
4. x2 – 10xy + c 81 4
121 4
36y2
25y2
Solution follows…

26. Completing the Square 7.2.1 Topic Independent Practice
Complete the square for each quadratic expression below.
5. x2 – 6x
6. a2 – 14a 7. b2 – 10b
8. x2 + 8xy 9. c2 – 12bc
10. x2 + 4xy
x2 – 6x + 9
a2 – 14a + 49
b2 – 10b + 25
x2 + 8xy + 16y2
c2 – 12bc + 36b2
x2 + 4xy + 4y2
Solution follows…

27. Completing the Square 7.2.1 Topic Independent Practice
Find the value of m and d in each of the following.
11. 5x2 – 40x + m = 5(x + d)2
12. 2x2 + 20x + m = 2(x + d) x2 – 6x + m = 3(x + d)2
14. 3x2 – 30x + m = 3(x + d) x2 + 24x + m = 4(x + d)2
16. 7x2 – 28x + m = 7(x + d)2
m = 80, d = –4
m = 50, d = 5
m = 3, d = –1
m = 75, d = –5
m = 36, d = 3
m = 28, d = –2
Solution follows…

28. Completing the Square 7.2.1 Topic Independent Practice
Add a term to convert each of the following into an expression of the form a(x + k)2.
17. 3x2 – 30x
18. 2x2 + 8x x2 – 48x
20. 5x x2 + 12
22. 36x
3(x – 5)2
2(x + 2)2
18(x – )2 4 3
5(x + 6)2 or 5(x – 6)2
27(x + )2 or 27(x – )2 2 3
36(x + )2 or 36(x – )2 7 3
Solution follows…

29. Completing the Square 7.2.1 Topic Independent Practice
23. The length of a rectangle is twice its width. If the area can be found by completing the square for (18x2 + 60x) ft2, find the width of the rectangle.
(3x + 5) ft
Solution follows…

30. Completing the Square 7.2.1 Topic Round Up
OK, so now you know how to add a number to a binomial to make a perfect square trinomial.
In the next Topic you’ll learn how to convert any quadratic expression into perfect square trinomial form — and then in Topic you’ll use this to solve quadratic equations.