Completing the Square
(For help, go to Lessons 9-4 and 9-7.)
Find each square.
1. (d – 4)2
2. (x + 11)2
3. (k – 8)2
5. t2 + 14t + 49
6. n2 – 18n + 81
Factor.
4. b2 + 10b + 25
Check Skills You’ll Need
10-5
Completing the Square
Solutions
1. (d – 4)2 = d2 – 2d(4) + 42 = d2 – 8d + 16
2. (x + 11)2 = x2 + 2x(11) + 112 = x2 + 22x + 121
3. (k – 8)2 = k2 – 2k(8) + 82 = k2 – 16k + 64
4. b2 + 10b + 25 = b2 + 2b(5) + 52 = (b + 5)2
5. t2 + 14t + 49 = t2 + 2t(7) + 72 = (t + 7)2
6. n2 – 18n + 81 = n2 – 2n(9) + 92 = (n – 9)2
10-5
Completing the Square
Find the value of c to complete the square for x2 – 16x + c.
The value of b in the expression x2 – 16x + c is –16.
The term to add to
x2
– 16x is
–16
2
2
or 64.
Quick Check
10-5
Completing the Square
First, write the left side of the equation as a perfect square.
X2 – 4x = 12
X2 – 4x + 4 = 12 + 4
Second, solve the equation by taking the square root of each side.
(x – 2)2 = V16
X – 2 = ±4
X = 4 + 2 and x = -4 + 2
X = 6 and -2
Completing the Square
Did you see that this can be factored using two binomials?.
X2 – 4x = 12
X2 – 4x – 12 = 0
(x – 6)( x+ 2) = 0
X = 6 and -2
Completing the Square
Solve the equation x2 + 5x = 50.
Step 1: Write the left side of x2 + 5x = 50 as a perfect square.
x2 + 5x = 50
x2 + 5x +
2
5
2
x+5
2
= 50 +
2
5
2
2
200
25
= 4 + 4
Add
5
2
2
, or 25 , to each side of the
4
equation.
Write x2 + 5x +
5
2
2
as a square.
Rewrite 50 as a fraction with
denominator 4.
x+5
2
2
=
225
4
10-5
Completing the Square
(continued)
Step 2: Solve the equation.
x+5
225
4
= ±
2
Find the square root of each side.
x + 5 = ± 15
2
x + 5 = 15
2
2
x=5
Simplify.
2
5
15
or
x+2 = – 2
or
x = –10
Write as two equations.
Solve for x.
Quick Check
10-5
Completing the Square
Solve x2 + 10x – 16 = 0 by completing the square. Round to
the nearest hundredth.
Step 1: Rewrite the equation in the form x2 + bx = c and complete
the square.
x2 + 10x – 16 = 0
x2 + 10x = 16
Add 16 to each side of the equation.
2
x2
+ 10x + 25 = 16 + 25
(x + 5)2 = 41
10
Add 2 , or 25, to each side of the equation.
Write x2 + 10x +25 as a square.
10-5
Completing the Square
(continued)
Step 2: Solve the equation.
x+5=±
x+5
41
Find the square root of each side.
x+5
± 6.40
6.40
or x + 5
x
6.40 – 5
or x
x
1.40
or
Use a calculator to find
–6.40
–6.40 – 5
x
–11.40
41
Write as two equations.
Subtract 5 from each side.
Simplify
Quick Check
10-5
Completing the Square
ALGEBRA 1 LESSON 10-5
Suppose you wish to section off a soccer field as shown in
the diagram to run a variety of practice drills. If the area of the field is
6000 yd2, what is the value of x?
Define: width = x + 10 + 10 = x + 20
length = x + x + 10 + 10 = 2x + 20
Relate: length  width = area
Write:
(2x + 20)(x + 20) = 6000
2x2 + 60x + 400 = 6000
Step 1: Rewrite the equation in the form x2 + bx = c.
2x2 + 60x + 400 = 6000
2x2 + 60x = 5600
x2 + 30x = 2800
Subtract 400 from each side.
Divide each side by 2.
10-5
Completing the Square
(continued)
Step 2: Complete the square.
2
x2 + 30x + 255 = 2800 + 225
(x + 15)2 = 3025
30
Add 2 , or 225, to each side.
Write x2 + 30x + 255 as a square.
Step 3: Solve each equation.
(x + 15) = ±
3025
x + 15 = ± 55
x + 15 = 55 or
x = 40 or
x + 15 = –55
x = –70
Take the square root of each side.
Use a calculator.
Use the positive answer for this problem.
Quick Check
The value of x is 40 yd.
10-5
Completing the Square
Solve each equation by completing the square. If necessary, round to the
nearest hundredth.
1. x2 + 14x = –43
–9.45, –4.55
2. 3x2 + 6x – 24 = 0
–4, 2
3. 4x2 + 16x + 8 = 40
–5.46, 1.46
10-5