Which of the following quadratic equations has the solutions $\frac{[11\pm \sqrt{(121+100)}]}{10}$?
a.
5$x$^{2} + 11$x$ - 5 = 0
b.
5$x$^{2} - 11$x$ - 5 = 0
c.
5$x$^{2} + 11$x$ + 5 = 0
d.
5$x$^{2} - 11$x$ + 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.]
ax^{2} + bx + c = 5x^{2} + (- 11)x + (- 5) [Substitute the values in the standard form.]
= 5x^{2} - 11x - 5 [Simplify.]
So, the quadratic equation is 5x^{2} - 11x - 5 = 0.
Correct answer : (2)
16.
Find the $x$-intercepts of the graph of $y$ = - $x$^{2} - 3$x$ + 10.
a.
- 2, - 5
b.
2, 5
c.
- 2, 5
d.
2, - 5
Solution:
The x-intercepts occur when y = 0.
y = - x^{2} - 3x + 10 [Original equation.]
0 = x^{2} + 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 = - x^{2} - 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 $h$ = - 16$t$^{2} + (- 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 = - 16t^{2} + (- 18)t + 9 [Original equation.]
0 = - 16t^{2} + (- 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 $h$ = - 16$t$^{2} + $\mathrm{vt}$ + $s$. ($s$ = 73.5 feet)
a.
3.06 seconds
b.
1.5 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 : (2)
19.
Find the $x$-intercepts of the graph of $y$ = $x$^{2} + 6$x$ - 55.
a.
- 11, - 5
b.
- 11, 5
c.
11, - 5
d.
11, 5
Solution:
y = x^{2} + 6x - 55 [Original equation.]
0 = x^{2} + 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 $h$ = - 16$t$^{2} + 729, where $t$ is the time in seconds.
a.
7.25
b.
6.50
c.
6.75
d.
7
Solution:
h = - 16t^{2} + 729 [Original equation.]
0 = - 16t^{2} + 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.]