﻿ Factoring Quadratic Trinomials Worksheet | Problems & Solutions

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1.
Factor: $r$2 + 16$r$ + 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).

2.
Factor: $r$2 - 7$r$ + 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).

3.
Factor: $z$2 - 16$z$ + 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).

4.
Factor: $d$2 + 24$d$ + 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).

5.
Factor: $n$2 + 22$n$ + 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)

6.
Factor: $p$2 - 24$p$ + 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)

7.
Factor: $z$2 - 17$z$ + 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).

8.
Factor: 4$x$2 + 29$x$ + 45
 a. (4$x$ - 9) ($x$ - 5) b. (4$x$ - 9) ($x$ + 5) c. (4$x$ + 9) ($x$ + 5) d. (4$x$ + 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)

9.
Factor: 2$a$2 + 13$a$ + 20
 a. (2$a$ - 5)($a$ - 4) b. (2$a$ + 5)($a$ + 4) c. (2$a$ + 2)($a$ + 10) d. (2$a$ + 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)

10.
Factor: 4$m$2 + 16$m$ + 16
 a. (4$m$ + 4)($m$ + 4) b. (2$m$ - 4)(2$m$ - 4) c. (2$m$ + 4)(2$m$ + 4) d. (2$m$ + 2)(2$m$ + 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)