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Factoring Quadratic Trinomials Worksheet

Factoring Quadratic Trinomials Worksheet
  • Page 1
 1.  
Factor: r2 + 16r + 63
a.
(r + 7)(r - 9)
b.
(r + 7)(r + 9)
c.
(r - 7)(r + 9)
d.
(r - 7)(r - 9)


Solution:

r2 + 16r + 63

The above equation is in the form of x2 + (a + b)x + ab = (x + a)(x + b).

The product of coefficent of r2 and the constant term are 1 × 63 = 63.
[ab = 63.]

In the given equation middle term is 16.
[a + b = 16.]

So the factors of 63 are 7 and 9 whose sum is 16.

So, r2 + 16r + 63 = (r + 7)(r + 9).


Correct answer : (2)
 2.  
Factor: r2 - 7r + 10
a.
(r - 2)(r - 5)
b.
(r - 1)(r - 10)
c.
(r - 1)(r + 10)
d.
(r - 2)(r + 5)


Solution:

r2 - 7r + 10

The above equation is in the form of x2 + (a + b)x + ab = (x + a)(x + b)

The product of coefficent of r2 and the constant term are 1 × 10 = 10.
[ab = 10.]

In the given equation middle term is - 7.
[a + b = - 7.]

So the factors of 10 are - 2 and - 5 whose sum is - 7.

Hence the factors of r2 - 7r + 10 are (r - 2) and (r - 5).


Correct answer : (1)
 3.  
Factor: z2 - 16z + 63
a.
(z + 1)(z + 63)
b.
(z - 7)(z - 9)
c.
(z + 7)(z + 9)
d.
(z - 1)(z - 63)


Solution:

z2 - 16z + 63

The two numbers whose product is the constant term 63 are 1 and 63; 7 and 9; - 1 and - 63; - 7 and - 9.

Write the possible factors and check the middle term.

Possible factors:                Middle term:

(z + 1)(z + 63)               63z + z = 64z

(z + 7)(z + 9)                 9z + 7z = 16z

(z - 1)(z - 63)                - 63z - z = - 64z

(z - 7)(z - 9)                  - 9z - 7z = - 16z
[Correct middle term.]

So, z2 - 16z + 63 = (z - 7)(z - 9).


Correct answer : (2)
 4.  
Factor: d2 + 24d + 80
a.
(d + 1)(d + 80)
b.
(d + 5)(d + 16)
c.
(d + 2)(d + 40)
d.
(d + 20)(d + 4)


Solution:

d2 + 24d + 80

The two numbers whose product is the constant term 80 are 1 and 80; 2 and 40; 4 and 20; 5 and 16; 8 and10.

Write the possible factors and check the middle term.

Possible factors:                 Middle term:

(d + 1)(d + 80)                 80d + d = 81d

(d + 2)(d + 40)                 40d + 2d = 42d

(d + 4)(d + 20)                 20d + 4d = 24d
[Correct middle term.]

(d + 5)(d + 16)                 16d + 5d = 21d

(d + 8)(d + 10)                 10d + 8d = 18d

So, d2 + 24d + 80 = (d + 4)(d + 20).


Correct answer : (4)
 5.  
Factor: n2 + 22n + 96
a.
(n + 8) (n + 12)
b.
(n + 6) (n + 16)
c.
(n + 2) (n + 48)
d.
(n + 4) (n + 24)


Solution:

n2 + 22n + 96

The two numbers whose product is the constant term 96 are 1 and 96; 2 and 48; 3 and 32; 4 and 24; 6 and 16; 8 and 12.

Write the possible factors and check the middle term.

Possible factors:          Middle term:

(n + 1) (n + 96)          96n + n = 97n

(n + 2) (n + 48)          48n + 2n = 50n

(n + 3) (n + 32)          32n + 3n = 35n

(n + 4) (n + 24)         24n + 4n = 28n

(n + 6) (n + 16)          16n + 6n = 22n
[Correct middle term.]

(n + 8) (n + 12)          12n + 8n = 20n

So, n2 + 22n + 96 = (n + 6) (n + 16)


Correct answer : (2)
 6.  
Factor: p2 - 24p + 108
a.
(p - 6)(p - 18)
b.
(p + 4)(p + 27)
c.
(p + 2)(p + 54)
d.
(p - 6)(p + 18)


Solution:

p2 - 24p + 108

The two numbers whose product is the constant term 108 are 1 and 108; 2 and 54; 3 and 36; 4 and 27; 6 and18; 9 and12.

Write the possible factors and check the middle term.

Possible factors:                 Middle term:

(p + 1)(p + 108)               108p + p = 109p

(p + 2)(p + 54)                 54p + 2p = 56p

(p + 3)(p + 36)                 36p + 3p = 39p

(p + 4)(p + 27)                 27p + 4p = 31p

(p + 6)(p + 18)                 18p + 6p = 24p

(p - 6)(p - 18)                 - 18p - 6p = - 24p
[Correct middle term.]

(p + 9)(n + 12)                 12p + 9p = 21p

So, p2 - 24p + 108 = (p - 6)(p - 18)


Correct answer : (1)
 7.  
Factor: z2 - 17z + 60
a.
(z - 5)(z - 12)
b.
(z - 4)(z - 15)
c.
(z - 2)(z - 30)
d.
(z - 5)(z + 12)


Solution:

z2 - 17z + 60

The two numbers whose product is the constant term 60 are 1 and 60; 2 and 30; 3 and 20; 4 and 15; 5 and 12; 6 and10.

Write the possible factors and check the middle term.

Possible factors:                 Middle term:

(z + 1)(z + 60)                 60z + z = 61z

(z + 2)(z + 30)                 30z + 2z = 32z

(z + 3)(z + 20)                 20z + 3z = 23z

(z + 4)(z + 15)                 15z + 4z = 19z

(z + 5)(z + 12)                 12z + 5z = 17z

(z - 5)(z - 12)                 - 12z - 5z = - 17z
[Correct middle term.]

(z + 6)(z + 10)                 10z + 6z = 16z

So, z2 - 17z + 60 = (z - 5)(z - 12).


Correct answer : (1)
 8.  
Factor: 4x2 + 29x + 45
a.
(4x - 9) (x - 5)
b.
(4x - 9) (x + 5)
c.
(4x + 9) (x + 5)
d.
(4x + 9) (x - 5)


Solution:

4x2 + 29x + 45

The product of coefficient of x2 and the constant term are 4 × 45 = 180

We have to find the factors whose product is 180 and sum is 29.

So the factors of 180 are 20 and 9.

4x2 + 29x + 45 = 4x2 + 20x + 9x + 45

= 4x(x + 5) + 9(x + 5)

= (4x + 9) (x + 5)
[Take out the common factors.]

So, 4x2 + 29x + 45 = (4x + 9) (x + 5)


Correct answer : (3)
 9.  
Factor: 2a2 + 13a + 20
a.
(2a - 5)(a - 4)
b.
(2a + 5)(a + 4)
c.
(2a + 2)(a + 10)
d.
(2a + 1)(a + 20)


Solution:

2a2 + 13a + 20

The two numbers whose product is the constant term 20 are 1 and 20; 2 and10; 4 and 5; 5 and 4; 20 and 1; 10 and 2.

Write the possible factors and check the middle term.

Possible factors:                 Middle term:

(2a + 1)(a + 20)                 40a + a = 41a

(2a + 20)(a + 1)                 2a + 20a = 22a

(2a + 2)(a + 10)                 20a + 2a = 22a

(2a + 10)(a + 2)                 4a + 10a = 14a

(2a + 4)(a + 5)                   10a + 4a = 14a

(2a + 5)(a + 4)                   8a + 5a = 13a
[Correct middle term.]

So, 2a2 + 13a + 20 = (2a + 5)(a + 4)


Correct answer : (2)
 10.  
Factor: 4m2 + 16m + 16
a.
(4m + 4)(m + 4)
b.
(2m - 4)(2m - 4)
c.
(2m + 4)(2m + 4)
d.
(2m + 2)(2m + 8)


Solution:

4m2 + 16m + 16

The two numbers whose product is the constant term 16 are 1 and 16; 2 and 8; 4 and 4.

Write the possible factors and check the middle term.

Possible factors:                     Middle term:

(2m + 1)(2m + 16)                32m + 2m = 34m

(2m + 2)(2m + 8)                 16m + 4m = 20m

(2m + 4)(2m + 4)                 8m + 8m = 16m
[Correct middle term.]

(4m + 2)(m + 8)                   2m + 32m = 34m

So, 4m2 + 16m + 16 = (2m + 4)(2m + 4)


Correct answer : (3)

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