Solved Examples and Worksheet for Factoring Binomials and Perfect Square Trinomials

Q1What term should be added to the expression x2 - 4x, to create a perfect square trinomial?
A. - 16
B. - 4
C. 4
D. 16

Solutions
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Step: 1
x2 - 4x
  [Original expression.]
Step: 2
Add to the expression x2 + bx, the square of half the coefficient of x, that is (b2)2 to create a perfect square trinomial.
Step: 3
The coefficient b = - 4, so add (b2)2 = (- 2)2, to the expression.
Step: 4
x2 - 4x + (- 2)2 = x2 - 4x + 4 = (x - 2)2 .
Step: 5
4 should be added to the expression x2 - 4x, to create a perfect square trinomial.
Correct Answer is :   4
Q2What term should be added to the expression, x2 + 14x to create a perfect square trinomial?

A. - 49
B. 49
C. 47
D. 50

Solutions
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Step: 1
x2 + 14x
  [Original expression.]
Step: 2
Add to the expression x2 + bx, the square of half the coefficient of x, that is (b2)² to create a perfect square trinomial.
Step: 3
The coefficient of x is 14, so add (142)2 = 72 to the expression.
Step: 4
x2 + 14x + (7)2 = x2 + 14x + 49 = (x + 7)2
Step: 5
49 should be added to the expression x2 + 14x, to create a perfect square trinomial.
Correct Answer is :   49
Q3Express as a perfect square.
u2 + 12u + 36

A. (u + 6)2
B. u(u + 6)
C. (u + 8)2
D. (u - 6)2

Solutions
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Step: 1
u2 + 12u + 36
  [Given expression.]
Step: 2
= u2 + 2(u)(6) +62
  [Write as (u2 + 2 u s +s2).]
Step: 3
= (u + 6)2
  [Write as (u + s)2.]
Correct Answer is :   (u + 6)2
Q4Express as a perfect square.
k2v2 + 6kv + 9

A. (kv + 3.2)2
B. (kv + 3)2
C. (kv - 2.9)2
D. (kv - 3)2

Solutions
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Step: 1
k2v2 +6kv + 9
  [Given expression.]
Step: 2
= (kv)2 + 2(kv)(3) +(3)2
  [Write as (a2 + 2ab +b2).]
Step: 3
= (kv + 3)2
  [Write as (a + b)2.]
Correct Answer is :    (kv + 3)2
Q5Express as a perfect square.
a2 - 18a + 81
A. (a + 10)2
B. (a + 9)2
C. (a - 9)2
D. (a - 11)2

Solutions
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Step: 1
a2 - 18a + 81
  [Given expression.]
Step: 2
= a2 - 2(a)(9) +92
  [Write as (a2 - 2ab +b2).]
Step: 3
= (a - 9)2
  [Write as (a - b)2.]
Correct Answer is :    (a - 9)2
Q6Express as a perfect square.
t2-10t4+2516
A. (t -15)2
B. (t +14)2
C. (t +54)2
D. (t -54)2

Solutions
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Step: 1
t2-10t4+2516
  [Given expression.]
Step: 2
= t2-2(t)(5)4+(54)2
  [Write as (a2-2ab+b2).]
Step: 3
= (t -54)2
  [Write as (a - b)2.]
Correct Answer is :    (t -54)2
Q7Express as a perfect square.
1w2+48w+576
A. (1w - 25)2
B. (1w + 25)2
C. (1w - 24)2
D. (1w + 24)2

Solutions
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Step: 1
1w2 +48w + 576
  [Original expression.]
Step: 2
Let 1w = a then 1w2 +48w + 576 = a2 + 48a + 576
Step: 3
= a2+2(a)(24)+242
  [Write as (a2+2ab+b2).]
Step: 4
= (a + 24)2
  [Write as (a + b)2.]
Step: 5
= (1w + 24)2
  [Replace a with 1w.]
Correct Answer is :    (1w + 24)2
Q8Express as a perfect square.
s2 - 10s + 25
A. (s - 7)2
B. (s - 5)2
C. (s + 6)2
D. (s + 5)2

Solutions
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Step: 1
s2 - 10s + 25
  [Original expression.]
Step: 2
= s2-2(s)(5)+52
  [Write as (a2-2ab+b2).]
Step: 3
= (s - 5)2
  [Write as (a - b)2.]
Correct Answer is :    (s - 5)2
Q9Factor:
16y2 - 36b2
A. (16y - b)(y - 9b)
B. (4y - 3b)2
C. (4y + 6b)(4y - 6b)
D. (4y + 3b) 2

Solutions
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Step: 1
16y2 - 36b2
Step: 2
= (4y)2 - (6b)2
  [Write as a2 - b2, a = 4y, b = 6b.]
Step: 3
= (4y + 6b)(4y - 6b)
  [a2 - b2 = (a + b) (a - b).]
Correct Answer is :   (4y + 6b)(4y - 6b)
Q10Factor:
16a2 - 169x2
A. (16a - x)(a - 9x)
B. (4a - 3x)2
C. (4a + 3x) 2
D. (4a + 13x)(4a - 13x)

Solutions
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Step: 1
16a2 - 169x2
Step: 2
= (4a)2 - (13x)2
  [Write as a2 - b2, a = 4a, b = 13x.]
Step: 3
= (4a + 13x)(4a - 13x)
  [a2 - b2 = (a + b) (a - b).]
Correct Answer is :   (4a + 13x)(4a - 13x)
Q11Factor:
64y2 - 289a2

A. (8y + 17a)(8y - 17a)
B. (8y + 17a) 2
C. (64y - a)(y - 17a)
D. (8y - 17a)2

Solutions
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Step: 1
64y2 - 289a2
Step: 2
= (8y)2 - (17a)2
  [Write as a2 - b2, a = 8y, b = 17a.]
Step: 3
= (8y + 17a)(8y - 17a)
  [a2 - b2 = (a + b) (a - b).]
Correct Answer is :   (8y + 17a)(8y - 17a)
Q12Factor:
x2 - 16225y2

A. (x - 415y)2
B. (x + 415y)(x - 415y)
C. (x + 16225y)(x - 16225y)
D. (x + 415y)2

Solutions
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Step: 1
x2 - 16225y2
Step: 2
= (x)2 - (415y)2
  [Write as a2 - b2, a = x, b = 415y.]
Step: 3
= (x + 415y)(x - 415y)
  [a2 - b2 = (a + b) (a - b).]
Correct Answer is :   (x + 415y)(x - 415y)
Q13Factor:
x2 - 64z2
A. (x + 8z)(x - 8z)
B. x(x - 8z)
C. (x - 8z)2
D. 8z(x - 8z)

Solutions
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Step: 1
x2 - 64z2
Step: 2
= (x)2 - (8z)2
  [Write as a2 - b2, a = x, b = 8z.]
Step: 3
= (x + 8z)(x - 8z)
  [a2 - b2 = (a + b) (a - b).]
Correct Answer is :    (x + 8z)(x - 8z)
Q14Factor:
x2 - 49z2
A. (x + 7z)(x - 7z)
B. x(x - 7z)
C. 7z(x - 7z)
D. (x - 7z)2

Solutions
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Step: 1
x2 - 49z2
Step: 2
= (x)2 - (7z)2
  [Write as a2 - b2, a = x, b = 7z.]
Step: 3
= (x + 7z)(x - 7z)
  [a2 - b2 = (a + b) (a - b).]
Correct Answer is :    (x + 7z)(x - 7z)
Q15Factor: 169g2 - 49h2

A. 13g2 - 7h2
B. (169g - 7h)(169g + 7h)
C. (13g - 7h)2
D. (13g + 7h)(13g - 7h)

Solutions
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Step: 1
169g2 - 49h2
  [Oringinal Expression.]
Step: 2
(13g)2 - (7h)2
  [Write as a2 - b2, where a = 13g, b= 7h.]
Step: 3
=(13g + 7h)(13g - 7h)
  [Since a2 - b2 = (a + b)(a - b).]
Step: 4
Therefore 169g2 - 49h2 = (13g + 7h)(13g - 7h).
Correct Answer is :   (13g + 7h)(13g - 7h)

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