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Quadratic Formula Worksheet - Page 2

Quadratic Formula Worksheet
  • Page 2
 11.  
Rewrite the equation 3h22 + 5 = h2 + (5h4), into the standard form and identify the values of a, b and c.
a.
a = 1 2, b = 5 4 and c = 5
b.
a = - 1 , b = - 5 and c = - 5
c.
a = 1 2, b = - 5 4 and c = - 5
d.
a = 1 2, b = - 5 4 and c = 5


Solution:

3h22 + 5 = h2 + (5h4)
[Original equation.]

(h22) + 5 = (5h4)
[Subtract h2 from each side.]

(h22) - (5h4) + 5 = 0
[Rewrite the equation in the standard form.]

a = 1 / 2, b = - 5 / 4 and c = 5
[Compare the equation with the standard form, ax2 + bx + c = 0.]


Correct answer : (4)
 12.  
Find the values of 7j, in the equation 14 j 2 - 1 = 12 j + 1.
a.
- 28, 14
b.
28, 14
c.
28, - 14
d.
- 28, - 14


Solution:

14 j 2 - 1 = 12 j + 1
[Original equation.]

14 j 2 - 1 - 12 j = 1
[Subtract 12 j from each side.]

14 j 2 - 2 - 12 j = 0
[Subtract 1 from each side.]

j2 - 8 - 2j = 0
[Multiply the equation with 4.]

j2 - 2j - 8 = 0
[Rewrite the equation in the standard form as ax2 + bx + c = 0.]

j = 4 or - 2
[Solve for j.]

7j = 7 × 4 = 28 or 7 × - 2 = - 14
[Substitute j and simplify.]


Correct answer : (3)
 13.  
Solve the equation 1 3k2 - 3k + 6 = 0 by using the quadratic formula.
a.
k = 3, 6
b.
k = - 3, - 6
c.
k = 9, 18
d.
k = 3 ±152


Solution:

1 / 3k2 - 3k + 6 = 0
[Original equation.]

k2 - 9k + 18 = 0
[Multiply the equation by 3.]

k = -(- 9) ±[(- 9)² - 4(1)(18)]2(1)
[Substitute a = 1, b = - 9 and c = 18 in the quadratic formula.]

k = 9 ±92
[Simplify inside the radical.]

k = (9 + 3)2, (9 - 3)2
[Write ± as two separate equations.]

k = 6, 3
[Simplify.]


Correct answer : (1)
 14.  
Compare 11 12n2 - 3 = 1 3n - 1 with the standard form and find the value of b2 - 4ac.
a.
- 1072
b.
1040
c.
1072
d.
-1040


Solution:

11 / 12n2 - 3 = 1 / 3n - 1
[Original equation.]

11 / 12n2 - 3 - 1 / 3n = - 1
[Subtract 1 / 3n from each side.]

11 / 12n2 - 1 / 3n - 2 = 0
[Add 1 to both sides.]

11n2 - 4n - 24 = 0
[Multiply the equation by 12.]

a = 11, b = - 4 and c = - 24
[Compare the equation with the standard form ax2 + bx + c = 0 and find the values.]

b2 - 4ac = (- 4)2 - 4(11)(- 24)
[Substitute the values.]

= 1072
[Simplify.]


Correct answer : (3)
 15.  
Which of the following quadratic equations has the solutions [11±(121+100)]10?
a.
5x2 + 11x - 5 = 0
b.
5x2 - 11x - 5 = 0
c.
5x2 + 11x + 5 = 0
d.
5x2 - 11x + 5 = 0


Solution:

Compare [11±(121+100)]10 with [-b±(b2-4ac)](2a)

2a = 10. So, a = 5.

- b = 11. So, b = - 11.

- 4ac = 100. So, c = - 5.
[Substitute a = 5 and simplify.]

ax2 + bx + c = 5x2 + (- 11)x + (- 5)
[Substitute the values in the standard form.]

= 5x2 - 11x - 5
[Simplify.]

So, the quadratic equation is 5x2 - 11x - 5 = 0.


Correct answer : (2)
 16.  
Find the x-intercepts of the graph of y = - x2 - 3x + 10.
a.
- 2, - 5
b.
2, 5
c.
- 2, 5
d.
2, - 5


Solution:

The x-intercepts occur when y = 0.

y = - x2 - 3x + 10
[Original equation.]

0 = x2 + 3x - 10
[Substitute y = 0 and write the equation in the standard form.]

x = {- 3 ±[(3)² - 4(1)(-10)]}2(1)
[Substitute a = 1, b = 2 and c = - 10 in the quadratic formula.]

= - 3±(9+40)2
[Simplify inside the radical.]

= - 3±492
[Add inside the radicals.]

= - 3±72
[Simplify the radical.]

= - 3 + 72 and - 3  -  72
[Write the expression as two separate terms.]

= 2 and - 5
[Simplify.]

So, the x-intercepts of the graph of y = - x2 - 3x + 10 are 2 and - 5.
[Simplify.]


Correct answer : (4)
 17.  
Diane dives into a pool from the diving board, which was 9 feet high from the water. She dives with an initial downward velocity of - 18 feet per second. If the equation to model the height of the dive is h = - 16t2 + (- 18)t + 9, then find the time in seconds taken by Diane to reach the water level.
a.
5.37
b.
0.74
c.
1.37
d.
0.37


Solution:

h = - 16t2 + (- 18)t + 9
[Original equation.]

0 = - 16t2 + (- 18t) + 9
[Replace h with 0, as the height is zero at the water level.]

t = {-(-18)±[(-18)2-4(-16)(9)]}[2(-16)]
[Substitute a = - 16, b = - 18 and c = 9 in the quadratic formula.]

t = 18±324+576-32
[Simplify.]

t = 18±900-32
[Simplify inside the radical.]

t = 18±30 / -32 = -1.50, 0.37
[Simplify the radical.]

t = 0.37
[Since t represents time, consider the positive integer.]


Correct answer : (4)
 18.  
Tony stands on a bridge 73.5 feet above the ground holding an apple. He throws it with an initial downward velocity of - 25 feet per second. How long will it take for the apple to reach the ground, if the vertical motion is given by the equation h = - 16t2 + vt + s.
(s = 73.5 feet)
a.
3.06 seconds
b.
1.5 seconds
c.
2 seconds
d.
2.5 seconds


Solution:

h = - 16t2 + vt + s
[Original equation.]

0 = - 16t2 + vt + s
[h = 0 for ground level.]

0 = - 16t2 - 25t + 73.5
[Replace v with - 25 and s = 73.5.]

t = [-(-25)±[(-25)2-4(-16)(73.5)]]2(-16)
[Substitute the values of a = - 16, b = - 25 and c = 73.5 in the quadratic formula.]

= [25±(625+4704)]-32
[Evaluate the power and multiply.]

= 25±5329-32
[Add within the grouping symbols.]

= 25±73-32
[Find the square root.]

t = - 3.0625 or 1.5
[Simplify.]

The apple will reach the ground about 1.5 seconds after it was thrown.


Correct answer : (2)
 19.  
Find the x-intercepts of the graph of y = x2 + 6x - 55.
a.
- 11, - 5
b.
- 11, 5
c.
11, - 5
d.
11, 5


Solution:

y = x2 + 6x - 55
[Original equation.]

0 = x2 + 6x - 55
[Substitute y = 0 to find the x-intercepts.]

x = (-6)±36+2202
[Substitute the values of a, b and c in the quadratic formula.]

= - 6±2562
[Simplify inside the radical symbol.]

= - 6±162
[Simplify the radical.]

= - 11 and 5
[Separate the terms and simplify.]

So, the x-intercepts are - 11 and 5.


Correct answer : (2)
 20.  
Frank jumped from a bungee tower, which was 729 feet high. Find the time taken by him to reach the ground, if the equation that models his height is h = - 16t2 + 729, where t is the time in seconds.
a.
7.25
b.
6.50
c.
6.75
d.
7


Solution:

h = - 16t2 + 729
[Original equation.]

0 = - 16t2 + 729
[Replace h with 0, as the height is zero at the ground level.]

t = {-(-0)±[(-0)2-4(-16)(729)]}[2(-16)]
[Substitute the values in the quadratic formula: a = - 16, b = 0 and c = 729.]

= 0±(0+46656)-32
[Simplify.]

= 0±46656-32
[Simplify inside the radical.]

= 0±216 / -32
[Simplify.]

= -216 / -32 = 6.75
[Since t represents time, use the positive solution.]


Correct answer : (3)

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