Question 1

Solve the quadratic equation using the method that seems most appropriate to you. $x ^ { 4 } - 153 x ^ { 2 } + 450 = 0$
A) $\{ \sqrt { 3 } , 5 \sqrt { 6 } \}$
B) $\{ \pm \sqrt { 3 } , 5 \sqrt { 6 } \}$
C) $\{ \pm \sqrt { 3 } , \pm 5 \sqrt { 6 } \}$
D) $\{ - \sqrt { 3 } , \pm 5 \sqrt { 6 } \}$
E) $\{ \sqrt { 3 } , \pm 5 \sqrt { 6 } \}$

Accepted Answer

This equation can be solved by substituting $y = x^2$, turning it into a quadratic equation $y^2 - 153y + 450 = 0$. Solving this equation yields $y = 9$ and $y = 50$, which correspond to $x^2 = 9$ and $x^2 = 50$. Therefore, the solutions for $x$ are $\pm 3$ and $\pm \sqrt{50}$, or more neatly, $\pm \sqrt{3}$ and $\pm 5\sqrt{6}$.

Question 2

Solve the quadratic equation using the method that seems most appropriate to you. $189 - 24 x - 5 x ^ { 2 } = 0$
A) $\left\{ 18 , - \frac { 42 } { 5 } \right\}$
B) $\left\{ - 9 , - \frac { 21 } { 5 } \right\}$
C) $\left\{ 9 , - \frac { 21 } { 5 } \right\}$
D) $\left\{ - 9 , \frac { 21 } { 5 } \right\}$
E) $\left\{ - 18 , \frac { 42 } { 5 } \right\}$

Accepted Answer

The answer of Solve the quadratic equation using the method...

Question 3

Solve the quadratic equation using the method that seems most appropriate to you. $x ^ { 2 } - 10 x + 17 = 0$
A) $\{ 10 \pm 4 \sqrt { 2 } \}$
B) $\{ 10 + 4 \sqrt { 2 } \}$
C) $\{ 10 - 4 \sqrt { 2 } \}$
D) $\{ 5 \pm 2 \sqrt { 2 } \}$
E) $\{ 5 - 2 \sqrt { 2 } \}$

Accepted Answer

The answer of Solve the quadratic equation using the method...

Question 4

Larry drove 336 miles in 1 hour more than it took Terrell to drive 324 miles. Terrell drove at an average rate of 6 miles per hour faster than Larry. How fast did Larry travel? __________ miles per hour

Accepted Answer

The answer of Larry drove 336 miles in 1 hour...

Question 5

Larry drove 360 miles in 2 hour more than it took Terrell to drive 300 miles. Terrell drove at an average rate of 5 miles per hour faster than Larry. How fast did Larry travel?
A) 42 mph
B) 49 mph
C) 46 mph
D) 45 mph
E) 40 mph

Accepted Answer

Larry's speed can be found by setting up equations based on the distance = rate × time formula. Let $x$ be Larry's speed in mph. Then Terrell's speed is $x + 5$ mph. If $t$ is the time it took Terrell to drive 300 miles, then Larry drove for $t + 2$ hours. The equations are $360 = x(t + 2)$ and $300 = (x + 5)t$. Solving these equations simultaneously, we find that Larry's speed $x = 45$ mph.

Question 6

Solve the quadratic equation using the method that seems most appropriate to you. $n + \frac { 6 } { n } = \frac { 29 } { 3 }$
A) $\left\{ - 9 , - \frac { 2 } { 3 } \right\}$
B) $\left\{ - 9 , \frac { 2 } { 3 } \right\}$
C) $\left\{ \frac { 2 } { 3 } , 9 \right\}$
D) $\left\{ - \frac { 2 } { 3 } , 9 \right\}$

Accepted Answer

Rearranging the equation to the form $n^2-\frac{29}{3}n+6=0$, we can solve it using the quadratic formula:
$$n=\frac{-(-\frac{29}{3})\pm\sqrt{(-\frac{29}{3})^2-4(1)(6)}}{2(1)}=\frac{29\pm\sqrt{449}}{6}$$
Therefore, the solutions are approximately $2.304$ and $9.021$, which matches choice C.

Question 7

Solve the quadratic equation using the method that seems most appropriate to you. $\frac { 49 } { x + 12 } - \frac { 19 } { x } = - 6$
A) { 38, 4 }
B) { 19, 2 }
C) { $-$ 38, $-$ 4 }
D) { 38, $-$ 4 }
E) { $-$ 19, 2 }

Accepted Answer

The answer of Solve the quadratic equation using the method...

Question 8

Solve the quadratic equation using the method that seems most appropriate to you. $x ^ { 2 } - 4 x - 19 = 0$ Please enter your answer as two numbers, separated by a comma.

Accepted Answer

The answer of Solve the quadratic equation using the method...

Question 9

The formula $D = \frac { n ( n - 3 ) } { 2 }$ yields the number of diagonals, D , in a polygon of n sides. Find the number of sides of a polygon that has 90 diagonals. __________ sides

Accepted Answer

The answer of The formula \(D = \frac { n...

Question 10

Solve the quadratic equation using the method that seems most appropriate to you. $3 y ^ { 2 } - 2 y + 1 = 0$
A) $\{ 1 \pm i \sqrt { 2 } \}$
B) $\{ 2 \pm 2 i \sqrt { 2 } \}$
C) $\left\{ \frac { 2 \pm 2 i \sqrt { 2 } } { 3 } \right\}$
D) $\left\{ \frac { 1 \pm i \sqrt { 2 } } { 3 } \right\}$

Accepted Answer

We can use the quadratic formula to solve this equation:
$$y=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
where $a=3$, $b=-2$, and $c=1$ from the given equation. Substituting these values, we get:
$$y=\frac{-(-2)\pm\sqrt{(-2)^2-4(3)(1)}}{2(3)}$$
Simplifying:
$$y=\frac{2\pm\sqrt{4-12}}{6}$$
$$y=\frac{2\pm\sqrt{-8}}{6}$$
$$y=\frac{1\pm i\sqrt{2}}{3}$$
Thus, the correct choice is D.

Question 11

Solve the quadratic equation using the method that seems most appropriate to you. $9 x ^ { 4 } + 11 x ^ { 2 } - 48 = 0$
A) $\left\{ \pm i \sqrt { 3 } , \pm \frac { 4 } { 3 } \right\}$
B) $\left\{ i \sqrt { 3 } , \frac { 4 } { 3 } \right\}$
C) $\left\{ \pm i \sqrt { 3 } , \frac { 4 } { 3 } \right\}$
D) $\left\{ i \sqrt { 3 } , \pm \frac { 4 } { 3 } \right\}$

Accepted Answer

Let $y = x^2$, transforming the equation into a quadratic form: $9y^2 + 11y - 48 = 0$. Solving this quadratic equation yields $y = \frac{4}{3}$ and $y = -3$. Substituting back $x^2 = y$, we get $x^2 = \frac{4}{3}$ and $x^2 = -3$, leading to the solutions $\left\{ \pm \frac{4}{3}, \pm i\sqrt{3} \right\}$.

Question 12

A man did a job for $168. It took him 2 hours longer than he expected, and therefore he earned $2 per hour less than he anticipated. How long did he expect that it would take to do the job?
A) 17 hours
B) 9 hours
C) 14 hours
D) 12 hours
E) 7 hours

Accepted Answer

Let's call the expected number of hours "x". Then he actually worked for "x+2" hours, and earned "$2 less per hour" than he expected, which means he earned $(2 less per hour) = $(2-x) per hour. The total amount he earned is given as $168, so we can set up an equation:

(x+2)(2-x) = 168

Expanding this out and simplifying gives:

-x^2 + 2x + 4x - 4 = 168
-x^2 + 6x - 172 = 0

We can solve for x using the quadratic formula:

x = (-6 ± sqrt(6^2 - 4(-1)(-172))) / (2(-1))
x = (-6 ± sqrt(1508)) / 2
x = (-6 ± 38.84) / 2

We're only interested in the positive solution, so:

x = (-6 + 38.84) / 2
x = 16.92

Therefore, he expected it to take about 16.92 hours to do the job, which we can round to 17 hours. The closest answer choice is (A) 17 hours, so that is the best choice.

Question 13

Find two consecutive whole numbers such that the sum of their squares is 25.
A) 6 and 1
B) 5 and 2
C) -1 and 8
D) 3 and 4
E) 1 and 6

Accepted Answer

The correct numbers are 3 and 4 because $3^2 + 4^2 = 9 + 16 = 25$.

Question 14

Suppose that the sum of two whole numbers is 13 and the sum of their reciprocals is $\frac { 13 } { 40 }$ Find the numbers.
A) 8 and 7
B) 7 and 6
C) 5 and 8
D) 6 and 8

Accepted Answer

Let the numbers be x and y. Then x + y = 13 and 1/x + 1/y = 13/40. Solving these equations, we find x = 5 and y = 8.

Question 15

The length of a rectangular floor is 8 meter less than twice its width. If a diagonal of the rectangle is 20 meters, find the length and width of the floor.
A) width is 11 m, length is 17 m
B) width is 10 m, length is 15 m
C) width is 24 m, length is 32 m
D) width is 12 m, length is 16 m
E) width is 14 m, length is 18 m

Accepted Answer

The answer of The length of a rectangular floor is...

Question 16

The formula $D = \frac { n ( n - 3 ) } { 2 }$ yields the number of diagonals, D , in a polygon of n sides. Find the number of sides of a polygon that has 35 diagonals.
A) 11 sides
B) 12 sides
C) 8 sides
D) 10 sides
E) 9 sides

Accepted Answer

We can use the given formula and solve for n:
\begin{align*}
D &= \frac{n(n-3)}{2}\
35 &= \frac{n(n-3)}{2}\
70 &= n(n-3) \
0 &= n^2 - 3n - 70 \
0 &= (n-10)(n+7) \
\end{align*}
Since we are dealing with the number of sides of a polygon, we can safely ignore the negative solution $n=-7$. Thus, the only valid solution is $n=10$, so the polygon has 10 sides. Therefore, the answer is D) 10 sides.

Question 17

Find two consecutive whole numbers such that the sum of their squares is 41.
A) 7 and 2
B) 6 and 3
C) 2 and 7
D) 4 and 5
E) 0 and 9

Accepted Answer

The correct consecutive whole numbers are 4 and 5, since $4^2 + 5^2 = 16 + 25 = 41$.

Question 18

Find two numbers such that their sum is 14 and their product is 48.
A) 8 and 5
B) 2 and 11
C) 4 and 9
D) 6 and 8
E) 9 and 4

Accepted Answer

The answer of Find two numbers such that their sum...

Question 19

A man did a job for $1,960. It took him 7 hours longer than he expected, and therefore he earned $5 per hour less than he anticipated. How long did he expect that it would take to do the job? __________ hours

Accepted Answer

The answer of A man did a job for $1,960....

Question 20

The perimeter of a rectangle is 30 inches and its area is 54 square inches. Find the length and width of the rectangle.
A) 5 in. by 8 in.
B) 8 in. by 9 in.
C) 9 in. by 10 in.
D) 6 in. by 9 in.
E) 6 in. by 11 in.

Accepted Answer

Let L be the length and W be the width. We have the perimeter equation: 2L + 2W = 30 (simplifies to L + W = 15). We also have the area equation: LW = 54. We can use substitution to solve for L or W in terms of the other variable. For example, we can solve for L: L = 15 - W. Substituting this into the area equation, we get W(15 - W) = 54. Simplifying and rearranging, we get W^2 - 15W + 54 = 0. This factors to (W - 6)(W - 9) = 0, so W = 6 or W = 9. If W = 6, then L = 15 - 6 = 9. If W = 9, then L = 15 - 9 = 6. Therefore, the possible dimensions are 6 inches by 9 inches or 9 inches by 6 inches. However, since length and width are interchangeable in a rectangle, both of these dimensions represent the same rectangle. Therefore, the best choice is D.

Quiz 11: Quadratic Equations and Inequalities

Solve the quadratic equation using the method that seems most appropriate to you. x4−153x2+450=0x ^ { 4 } - 153 x ^ { 2 } + 450 = 0x4−153x2+450=0

Solve the quadratic equation using the method that seems most appropriate to you. 189−24x−5x2=0189 - 24 x - 5 x ^ { 2 } = 0189−24x−5x2=0

Solve the quadratic equation using the method that seems most appropriate to you. x2−10x+17=0x ^ { 2 } - 10 x + 17 = 0x2−10x+17=0

Larry drove 336 miles in 1 hour more than it took Terrell to drive 324 miles. Terrell drove at an average rate of 6 miles per hour faster than Larry. How fast did Larry travel? __________ miles per hour

Larry drove 360 miles in 2 hour more than it took Terrell to drive 300 miles. Terrell drove at an average rate of 5 miles per hour faster than Larry. How fast did Larry travel?

Solve the quadratic equation using the method that seems most appropriate to you. n+6n=293n + \frac { 6 } { n } = \frac { 29 } { 3 }n+n6​=329​

Solve the quadratic equation using the method that seems most appropriate to you. 49x+12−19x=−6\frac { 49 } { x + 12 } - \frac { 19 } { x } = - 6x+1249​−x19​=−6

Solve the quadratic equation using the method that seems most appropriate to you. x2−4x−19=0x ^ { 2 } - 4 x - 19 = 0x2−4x−19=0 Please enter your answer as two numbers, separated by a comma.

The formula D=n(n−3)2D = \frac { n ( n - 3 ) } { 2 }D=2n(n−3)​ yields the number of diagonals, D , in a polygon of n sides. Find the number of sides of a polygon that has 90 diagonals. __________ sides

Solve the quadratic equation using the method that seems most appropriate to you. 3y2−2y+1=03 y ^ { 2 } - 2 y + 1 = 03y2−2y+1=0

Solve the quadratic equation using the method that seems most appropriate to you. 9x4+11x2−48=09 x ^ { 4 } + 11 x ^ { 2 } - 48 = 09x4+11x2−48=0

A man did a job for $168. It took him 2 hours longer than he expected, and therefore he earned $2 per hour less than he anticipated. How long did he expect that it would take to do the job?

Find two consecutive whole numbers such that the sum of their squares is 25.

Suppose that the sum of two whole numbers is 13 and the sum of their reciprocals is 1340\frac { 13 } { 40 }4013​ Find the numbers.

The length of a rectangular floor is 8 meter less than twice its width. If a diagonal of the rectangle is 20 meters, find the length and width of the floor.

The formula D=n(n−3) 2D = \frac { n ( n - 3 ) } { 2 }D=2n(n−3) ​ yields the number of diagonals, D , in a polygon of n sides. Find the number of sides of a polygon that has 35 diagonals.

Find two consecutive whole numbers such that the sum of their squares is 41.

Find two numbers such that their sum is 14 and their product is 48.

A man did a job for $1,960. It took him 7 hours longer than he expected, and therefore he earned $5 per hour less than he anticipated. How long did he expect that it would take to do the job? __________ hours

The perimeter of a rectangle is 30 inches and its area is 54 square inches. Find the length and width of the rectangle.

Solve the quadratic equation using the method that seems most appropriate to you. $x ^ { 4 } - 153 x ^ { 2 } + 450 = 0$

Solve the quadratic equation using the method that seems most appropriate to you. $189 - 24 x - 5 x ^ { 2 } = 0$

Solve the quadratic equation using the method that seems most appropriate to you. $x ^ { 2 } - 10 x + 17 = 0$

Solve the quadratic equation using the method that seems most appropriate to you. $n + \frac { 6 } { n } = \frac { 29 } { 3 }$

Solve the quadratic equation using the method that seems most appropriate to you. $\frac { 49 } { x + 12 } - \frac { 19 } { x } = - 6$

Solve the quadratic equation using the method that seems most appropriate to you. $x ^ { 2 } - 4 x - 19 = 0$ Please enter your answer as two numbers, separated by a comma.

The formula $D = \frac { n ( n - 3 ) } { 2 }$ yields the number of diagonals, D , in a polygon of n sides. Find the number of sides of a polygon that has 90 diagonals. __________ sides

Solve the quadratic equation using the method that seems most appropriate to you. $3 y ^ { 2 } - 2 y + 1 = 0$

Solve the quadratic equation using the method that seems most appropriate to you. $9 x ^ { 4 } + 11 x ^ { 2 } - 48 = 0$

Suppose that the sum of two whole numbers is 13 and the sum of their reciprocals is $\frac { 13 } { 40 }$ Find the numbers.

The formula $D = \frac { n ( n - 3 ) } { 2 }$ yields the number of diagonals, D , in a polygon of n sides. Find the number of sides of a polygon that has 35 diagonals.