Solving Quadratic Equations by Factoring Worksheet

Solving Quadratic Equations by Factoring Worksheet
• Page 1
1.
Solve the equation ($a$ - 5)($a$ + 5) = 0.
 a. 5 and - 5 b. 5 and 5 c. -5 and + 5 d. - 5 and 5

Solution:

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

(a - 5) = 0 or (a + 5) = 0
[Set each factor equal to zero.]

a = 5 or a = -5
[Solve for a.]

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

2.
What is the $x$-coordinate of the vertex of the graph of $y$ = 5(4$x$ + 9)($x$ + 3)?
 a. - $\frac{20}{7}$ b. - $\frac{8}{21}$ c. - $\frac{21}{8}$ d. - $\frac{22}{9}$

Solution:

y = 5(4x + 9)(x + 3)
[Original equation.]

0 = 5(4x + 9)(x + 3)
[Substitute y = 0.]

(4x + 9) = 0 or (x + 3) = 0
[Solve for x.]

The x-intercepts are -9 / 4 and -3.

The x-coordinate of the vertex is the average of the x-intercepts.

x = [(-94) + (-3)]/2 = -218

The x-coordinate of the vertex of the graph of y = 5(4x + 9)(x + 3) is Ã¢â‚¬â€œ21 / 8

3.
The entrance door of a prison is in an inverse U-shape, similar to a parabola. If the shape of the door is modeled by a function $y$ = (-4/32)(2$x$ + 4)(2$x$ - 4), then what is the width of the door at the base?
 a. 3 b. 4 c. 5 d. 6

Solution:

y = (-4/32)(2x + 4)(2x - 4)
[Original equation.]

0 = (-4/32)(2x - 4)(2x + 4)
[Substitute 0 for y to get x-intercepts.]

(2x - 4) = 0 or (2x + 4) = 0
[Set each factor equal to zero.]

x = 2 or x = - 2
[Solve for x.]

The width of the door at the base is 2 + 2 = 4 ft.
[Take positive value because the width cannot be negative.]

4.
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.

5.
Solve: ($b$ - 7)2 = 0
 a. 7 b. 12 c. -12 d. -7

Solution:

(b - 7)2 = 0
[Original equation.]

(b - 7)(b - 7) = 0
[Split into factors.]

b - 7 = 0
[Equate each factor to zero.]

b = 7
[Simplify.]

The solution of the equation (b - 7)2 = 0 is 7.

6.
Solve: (10$c$ - 9)2 = 0
 a. $\frac{9}{10}$ b. $\frac{10}{9}$ c. - $\frac{10}{9}$ d. - $\frac{9}{10}$

Solution:

(10c - 9)2 = 0
[Original equation.]

(10c - 9)(10c - 9) = 0
[Split into factors.]

(10c - 9) = 0
[Equate each factor to zero.]

10c = 9
[Simplify.]

c = 910
[Divide each side by 10.]

The solution of the equation (10c - 9)2 = 0 is 9 / 10.

7.
Solve: ($d$ - 5)(5$d$ - 11) = 0
 a. - 5 and - $\frac{11}{5}$ b. 5 and - $\frac{11}{5}$ c. 5 and $\frac{11}{5}$ d. -5 and $\frac{11}{5}$

Solution:

(d - 5)(5d - 11) = 0
[Original equation.]

(d - 5) = 0 or (5d - 11) = 0
[Equate each factor to zero.]

d = 5 or d = 115
[Simplify.]

The solutions of the equation (d - 5)(5d - 11) = 0 are 5 and 11 / 5.

8.
Solve: $f$($f$ + 3) = 0
 a. 3 and -3 b. 0 and 3 c. 5 and -5 d. 0 and -3

Solution:

f (f + 3) = 0
[Original equation.]

f = 0 or (f + 3) = 0
[Equate each factor to zero.]

f = 0 or f = -3
[Simplify.]

The solutions of the equation f(f + 3) = 0 are 0 and -3.

9.
Solve: ($g$ - 3)($g$ + 3)($g$ - 4) = 0
 a. 3 and -3 or -4 b. -3 and -3 or 4 c. 3, -3 and 4 d. 3 and 3 or -4

Solution:

(g - 3)(g + 3)(g - 4) = 0
[Original equation.]

(g - 3) = 0 or (g + 3) = 0 or (g - 4) = 0
[Equate each factor to zero.]

g = 3 or g = -3 or g = 4
[Simplify.]

The solutions of the equation (g - 3)(g + 3)(g - 4) = 0 are 3, -3 and 4.

10.
Solve: (4$h$ - 3)(5$h$ - 4)(6$h$ + 5) = 0
 a. -3/4, -4/5 and -5/6 b. 3/4, 4/5 and -5/6 c. -3/4, -4/5 and 5/6 d. 3/4, 4/5 and 5/6

Solution:

(4h - 3)(5h - 4)(6h + 5) = 0
[Original equation.]

(4h - 3) = 0 or (5h - 4) = 0 or (6h + 5) = 0
[Equate each factor to zero.]

4h = 3 or 5h = 4 or 6h = -5
[Simplify.]

h = 3 / 4 or h = 4 / 5 or h = -5 / 6
[Simplify to get the value of h.]

The solutions of the equation (4h - 3)(5h - 4)(6h + 5) = 0 are 3 / 4, 4 / 5 and - 5 / 6.