-4ac = 40. So, c = -2. [Substitute a = 5 and simplify.]
ax^{2} + bx + c = 5x^{2} + (-11)x + (-2) [Substitute the values in the standard form.]
= 5x^{2} - 11x - 2 [Simplify.]
So, the quadratic equation is 5x^{2} - 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 $h$ = -16$t$^{2} - 30$t$ + 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 = -16t^{2} - 30t + 124 [Original equation.]
0 = -16t^{2} - 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 $y$ = -$x$^{2} - 3$x$ + 18.
a.
3, -6
b.
3, 6
c.
-3, 6
d.
-3, -6
Solution:
The x-intercepts occur when y = 0.
y = -x^{2} - 3x + 18 [Original equation.]
0 = x^{2} + 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 $h$ = -16$t$^{2} +(-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 = -16t^{2} + (-12)t + 4 [Original equation.]
0 = -16t^{2} + (-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 $h$ = -16$t$^{2} + $v$$t$ + $s$?
a.
1.5 seconds
b.
3.06 seconds
c.
2 seconds
d.
2.5 seconds
Solution:
h = -16t^{2} + vt + s [Original equation.]
0 = -16t^{2} + vt + s [h = 0 for ground level.]
0 = -16t^{2} - 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 $y$ = $x$^{2} + 5$x$ - 24.
a.
-8, -3
b.
8, -3
c.
-8, 3
d.
None of the above
Solution:
y = x^{2} + 5x - 24 [Original equation.]
0 = x^{2} + 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 $h$ = -16$t$^{2} + 784, where $t$ is the time in seconds.
a.
6.75
b.
7.25
c.
7
d.
7.5
Solution:
h = -16t^{2} + 784 [Original equation.]
0 = -16t^{2} + 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 $h$ = -16$t$^{2} + 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 = -16t^{2} + 9 [Original equation.]
0 = -16t^{2} + 9 [Substitute 0 for h, since at ground level the height is zero.]
t = {-0 ± √[0^{2} - 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 $p$ = $s$^{2} + 2$s$ - 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 = s^{2} + 2s - 1520 [Original equation.]
0 = s^{2} + 2s - 1520 [Substitute 0 for p, since for the minimum price, the equation should be equal to zero.]
s = {-(2) ± √[2^{2} - 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.]