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Solving Quadratic Equations by Quadratic Formula Worksheet - Page 2

Solving Quadratic Equations by Quadratic Formula Worksheet
  • Page 2
 11.  
Compare 3 4n2 - 4 = 1 2n - 1 with the standard form and find the value of b2 - 4ac.
a.
148
b.
-148
c.
-140
d.
140


Solution:

34n2 - 4 = 12n - 1
[Original equation.]

34n2 - 4 - 12n = -1
[Subtract 1 / 2n from each side.]

34n2 - 12n - 3 = 0
[Add 1 to each side.]

3n2 - 2n - 12 = 0
[Multiply the equation by 4.]

a = 3, b = -2 and c = -12
[Compare the equation with the standard form ax2 + bx + c and find the values.]

b2 - 4ac = (-2)2 - 4(3)(-12)
[Substitute the values.]

= 148
[Simplify.]


Correct answer : (1)
 12.  
Which of the following quadratic equations has the solutions [11 ± √(121 + 40)] / 10?
a.
5x2 - 11x - 2 = 0
b.
5x2 + 11x - 2 = 0
c.
5x2 + 11x + 2 = 0
d.
None of the above


Solution:

Compare [11 ± √(121 + 40)] / 10 with [-b ± √(b2 - 4ac)]/(2a)

2a = 10. So, a = 5.

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

-4ac = 40. So, c = -2.
[Substitute a = 5 and simplify.]

ax2 + bx + c = 5x2 + (-11)x + (-2)
[Substitute the values in the standard form.]

= 5x2 - 11x - 2
[Simplify.]

So, the quadratic equation is 5x2 - 11x - 2 = 0.


Correct answer : (1)
 13.  
Jake throws a pen from the top of a building with an initial downward velocity of -30 feet per second. How long will the pen take to reach the ground, if the height of the pen from the ground is modeled by the equation h = -16t2 - 30t + 124, where h is the height of the pen and t is the time in seconds?
a.
124
b.
2
c.
-3
d.
30


Solution:

h = -16t2 - 30t + 124
[Original equation.]

0 = -16t2 - 30t + 124
[Height = 0, when the pen is on the ground.]

Compare the original equation with the standard form to get the values of a, b and c.

t = [-(-30) ± √[(-30)2 - 4(-16)(124)]] / [2(-16)]
[Substitute the values in the quadratic formula.]

t = [30 ± √(900 + 7936)]/(-32)
[Evaluate power and multiply.]

= [30 ± 94]/(-32)
[Simplify the radical.]

= - 3.875, 2
[Simplify.]

The ball reaches the ground after 2 seconds.
[Consider positive value as t represents time.]


Correct answer : (2)
 14.  
Find the x-intercepts of the graph of y = -x2 - 3x + 18.
a.
3, -6
b.
3, 6
c.
-3, 6
d.
-3, -6


Solution:

The x-intercepts occur when y = 0.

y = -x2 - 3x + 18
[Original equation.]

0 = x2 + 3x - 18
[Substitute y = 0 and write the equation in the standard form.]

x = {-3 ± √[(3)2 - 4(1)(-18)]} / 2(1)
[Substitute a = 1, b = 3 and c = -18 in the quadratic formula.]

= -3±√(9+72) / 2
[Simplify the squares.]

= -3±√81 / 2
[Add inside the radicals.]

= -3±9 / 2
[Simplify the radical.]

= -3+9 / 2 , -3-9 / 2
[Write ± as two separate terms.]

= 3, -6
[Simplify.]


Correct answer : (1)
 15.  
Olga dives into a pool from the diving board, which was 4 feet high from the water. She dives with an initial downward velocity of -12 feet per second. If the equation to model the height of the dive is h = -16t2 +(-12)t + 4, then find the time (in seconds) Olga takes to reach the water level.
a.
0.25
b.
2.25
c.
1.25
d.
None of the above


Solution:

h = -16t2 + (-12)t + 4
[Original equation.]

0 = -16t2 + (-12t) + 4
[Substitute 0 for h, since at water level the height is zero.]

t = {-(-12) ± √[(-12)2 - 4(-16)(4)]}/[2(-16)]
[Substitute a = -16, b = -12 and c = 4 in the quadratic formula.]

t = 12±√(144+256) / -32
[Simplify.]

t = 12±√400 / -32
[Simplify inside the radical.]

t = 12±20 / -32 = -1, 0.25
[Simplify the radical.]

t = 0.25
[Since t represents time, consider the positive integer.]


Correct answer : (1)
 16.  
Edward 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?
a.
1.5 seconds
b.
3.06 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 : (1)
 17.  
Find the x-intercepts of the graph of y = x2 + 5x - 24.
a.
-8, -3
b.
8, -3
c.
-8, 3
d.
None of the above


Solution:

y = x2 + 5x - 24
[Original equation.]

0 = x2 + 5x - 24
[Substitute y = 0 to find the x-intercepts.]

= -5±√(25+96) / 2
[Substitute the values of a, b and c in the quadratic formula.]

= -5±√121 / 2
[Simplify inside the radical symbol.]

= -5±11 / 2
[Evaluate the power.]

x = -8 or 3
[Simplify.]

So, the x-intercepts are -8 and 3.


Correct answer : (3)
 18.  
Sunny jumped from a bungee tower, which was 784 feet high. Find the time taken by him to reach the ground, if the equation that models his height is h = -16t2 + 784, where t is the time in seconds.
a.
6.75
b.
7.25
c.
7
d.
7.5


Solution:

h = -16t2 + 784
[Original equation.]

0 = -16t2 + 784
[Substitute 0 for h, since at ground level the height is zero.]

t = {-(-0) ± √[(-0)2 - 4(-16)(784)]}/[2(-16)]
[Substitute the values in the quadratic formula: a = -16, b = 0 and c = 784.]

= 0±√(0+50176)] / -32
[Simplify.]

= 0±√50176 / -32
[Simplify inside the radical.]

= 0±224 / -32
[Simplify.]

= -224 / -32 = 7
[Since t represents time, rounding t to a positive number.]


Correct answer : (3)
 19.  
A stone is dropped from a height of 9 feet above the ground. The height of the stone is modeled by the equation h = -16t2 + 9, where t is the time in seconds. Find the time taken for the stone to hit the ground.
a.
0.5
b.
1
c.
1.25
d.
0.75


Solution:

h = -16t2 + 9
[Original equation.]

0 = -16t2 + 9
[Substitute 0 for h, since at ground level the height is zero.]

t = {-0 ± √[02 - 4(-16)(9)]}/(2(-16))
[Substitute the values in the quadratic formula: a = -16, b = 0 and c = 9.]

= 0±√(0+576) / -32
[Simplify.]

0±√576 / -32
[Simplify inside the radical.]

= 0±24 / -32
[Simplify the radical.]

t = -24 / -32 = 0.75
[Since t represents time, rounding it to a positive number.]


Correct answer : (4)
 20.  
The cost price of cotton purchased by a textile company is given by the function p = s2 + 2s - 1520, where s is the number of bales purchased. How many bales of cotton are to be purchased for the cost price to be minimum?
a.
40
b.
38
c.
42
d.
44


Solution:

p = s2 + 2s - 1520
[Original equation.]

0 = s2 + 2s - 1520
[Substitute 0 for p, since for the minimum price, the equation should be equal to zero.]

s = {-(2) ± √[22 - 4(1)(-1520)]}/2(1)
[Substitute the values in the quadratic formula: a = 1, b = 2 and c = -1520.]

= -2±√(4+6080) / 2
[Simplify.]

= -2±√6084 / 2
[Simplify inside the radical.]

= -2±78 / 2
[Simplify the radical.]

s = 76 / 2 = 38
[Since s is a function of price, rounding it to a positive integer.]


Correct answer : (2)

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