# Solving Equations by Factoring Worksheet

Solving Equations by Factoring Worksheet
• Page 1
1.
Find the value of $x$, which satisfies the equation |$x$2 + 7$x$| + 2$x$2 - 6 = 0 if $x$2 + 7$x$ > 0.
 a. - $\frac{3}{2}$ b. 2 c. $\frac{2}{3}$ d. - 3

#### Solution:

|x2 + 7x| + 2x2 - 6 = 0 for x2 + 7x > 0

x2 + 7x + 2x2 - 6 = 0

3x2 + 7x - 6 = 0
[Simplify.]

(x + 3)(3x - 2) = 0
[Split into factors.]

x + 3 = 0 or 3x - 2 = 0
[Equate each factor to zero.]

x = - 3 or x = 2 / 3
[Simplify.]

Therefore, x = 2 / 3
[As x = - 3 does not satisfy the condition x2 + 7x > 0.]

2.
Andrew's garden is square shaped and Gary's garden is rectangular in shape. The length of the rectangular garden is longer by $a$ ft, and its width is $b$ ft shorter than the square garden. If both gardens have the same area, then find the dimensions of the rectangular garden. [$a$ = 9 ft and $b$ = 7 ft.]
 a. 72.0 ft × 40.5 ft b. 40.5 ft × 24.5 ft c. 40.5 ft × 38.5 ft d. 24.5 ft × 56.0 ft

#### Solution:

Let x be the edge length of the square garden.

If the dimensions of the square garden are x × x, then the dimensions of the rectangular garden are (x + a) ft × (x - b) ft.
That is (x + 9) ft × (x - 7) ft

Square garden area = x2 ft2

Area of the rectangular garden = (x + 9)(x - 7) ft2

If the areas of 2 gardens are equal, then x2 = (x + 9)(x - 7)

x2 = x2 + 9x - 7x - 63

2x - 63 = 0

x = 31.5

So, the dimensions of the new rectangular garden are 40.5 ft × 24.5 ft.
[Substitute x = 31.5 in (x + 9) and (x - 7).]

3.
Solve:
($e$ + 5)2 = 0
 a. 25 b. - 25 c. - 5

#### Solution:

(e + 5)2 = 0
[Original equation.]

(e + 5)(e + 5) = 0
[Split into factors.]

(e + 5) = 0
[Equate each factor to zero.]

e = - 5
[Solve for e.]

The solution of the equation (e + 5)2 = 0 is - 5.

4.
Solve = 0.
 a. - 2.8 b. 2.8 c. - 1.4 d. 1.4

#### Solution:

(a - 1.4)2 = 0
[Original equation.]

a - 1.4 = 0
[Set repeated factor equal to 0.]

a = 1.4
[Solve for a.]

The solution of the equation (a - 1.4)2 = 0 is 1.4 .

5.
Solve ($b$ - 9)($b$ - 13)2 = 0.
 a. - 9 and 13 b. 9 and 13 c. - 9 and - 13 d. 9 and - 13

#### Solution:

(b - 9)(b - 13)2 = 0
[Original equation.]

b - 9 = 0 or b - 13 = 0
[Equate each factor to zero.]

b = 9 or b = 13
[Solve for b.]

The solutions of the equation (b - 9)(b - 13)2 = 0 are 9 and 13.

6.
Solve (9$x$ - 20.7)(3$x$ + 10.8)2 = 0.
 a. 20.7 and - 10.8 b. 22.7 and 3.6 c. 2.3 and 13.8 d. 2.3 and - 3.6

#### Solution:

(9x - 20.7)(3x + 10.8)2 = 0
[Original equation.]

9x - 20.7 = 0 or 3x + 10.8 = 0
[Equate each factor to zero.]

x = 2.3 or x = - 3.6
[Solve for x.]

The solutions of the equation (9x - 20.7)(3x + 10.8)2 = 0 are 2.3 and - 3.6 .

7.
Solve ($y$ - $\frac{1}{2}$)($y$ + 5) = 0.
 a. - $\frac{1}{2}$ and - 5 b. - $\frac{1}{2}$ and 5 c. $\frac{1}{2}$ and 5 d. $\frac{1}{2}$ and - 5

#### Solution:

(y - 12)(y + 5) = 0
[Original equation.]

(y - 12) = 0 or (y + 5) = 0
[Equate each factor to zero.]

y = 12 or y = - 5
[Solve for y.]

The solutions of the equation (y - 12)(y + 5) = 0 are 12 and - 5.

8.
Solve ($u$ - $\frac{3}{4}$)(5$u$ + 8)($\frac{1}{5}u$ - 7) = 0.
 a. $\frac{3}{4}$, $\frac{8}{5}$ and 7 b. $\frac{3}{4}$, - $\frac{8}{5}$ and 35 c. - $\frac{3}{4}$, $\frac{8}{5}$ and 7 d. $\frac{3}{4}$, 3 and - 35

#### Solution:

(u - 34)(5u + 8)(15u - 7) = 0
[Original equation.]

(u - 34) = 0 or (5u + 8) = 0 or (15u - 7) = 0
[Equate each factor to zero.]

u = 34 or u = - 8 / 5 or u = 35
[Solve for u.]

The solutions of the equation (u - 34)(5u + 8)(15u - 7) = 0 are 34, - 8 / 5 and 35.

9.
Solve the equation ($a$ - 4)($a$ + 4) = 0.
 a. - 4 and 0 b. 4 and - 4 c. 4 and 4 d. - 4 and - 4

#### Solution:

(a - 4)(a + 4) = 0
[Original equation.]

(a - 4) = 0 or (a + 4) = 0
[Set each factor equal to zero.]

a = 4 or a = - 4
[Solve for a.]

The solutions for the equation (a - 4)(a + 4) = 0 are 4 and - 4.

10.
Solve:
$f$($f$ + 2) = 0
 a. 0 and - 2 b. 4 and - 4 c. 0 and 2 d. 2 and - 2

#### Solution:

f (f + 2) = 0
[Original equation.]

f = 0 or (f + 2) = 0
[Equate each factor to zero.]

f = 0 or f = - 2
[Simplify.]

The solutions of the equation f(f + 2) = 0 are 0 and - 2.