Question 1

Solve by using the quadratic formula.
-$7 x=\sqrt{\frac{49 x-6}{2}}$

A) $\left\{\frac{3}{14}\right\}$

B) $\left\{\frac{2}{7}\right\}$

C) $\left\{\frac{2}{7}, \frac{3}{14}\right\}$

D) $\left\{-\frac{2}{7}, \frac{3}{14}\right\}$

Accepted Answer

Squaring both sides of the equation gives $49x^2=49x-6$, which can be rewritten as $49x^2-49x+6=0$. Using the quadratic formula, we get $x=\frac{49\pm\sqrt{49^2-4(49)(6)}}{2(49)}=\frac{49\pm\sqrt{2401-1176}}{98}=\frac{49\pm\sqrt{1225}}{98}=\frac{49\pm35}{98}$. Therefore, $x=\frac{3}{14}$ or $x=\frac{2}{7}$.

Question 2

Decide whether the statement is true or false. If it is false, tell why.
-If $k$ is a positive number, then the equation $-x^{2}=-100 k$ has no real solutions.
A) True
B) False; the equation is equivalent to $x^{2}=100 k$, where $x^{2}$ and $100 k$ are positive, so it will have two real solutions
C) False; the equation is equivalent to $x^{2}=100 k$, where $x^{2}$ and $100 k$ are positive, so the equation will have one real number solution

Accepted Answer

The equation $-x^{2}=-100 k$ simplifies to $x^{2}=100 k$. Since $k$ is positive, $100 k$ is also positive, meaning $x^{2}$ is positive. This results in two real solutions for $x$, as a positive square root has both a positive and negative solution.

Question 3

What is the discriminant for $4 x^{2}-12 x+9=0$ ? How many and what type of solutions does this equation have?
A) Discriminant: 64; Two rational solutions
B) Discriminant: 63; Two irrational solutions
C) Discriminant: 0; One rational solution
D) Discriminant: -64; Two nonreal complex solutions

Accepted Answer

The discriminant for the quadratic equation $ax^2 + bx + c = 0$ is given by $b^2 - 4ac$. For $4x^2 - 12x + 9 = 0$, $a = 4$, $b = -12$, and $c = 9$. The discriminant is $(-12)^2 - 4(4)(9) = 144 - 144 = 0$. A discriminant of 0 indicates one rational solution.

Question 4

Solve by any method.
-$1-\frac{7}{x}-\frac{60}{x^{2}}=0$
A) $\{-12,5\}$
B) $\{12,5\}$
C) $\{12,-5\}$
D) $\{-12,-5\}$

Accepted Answer

The answer of Solve by any method.
-$1-\frac{7}{x}-\frac{60}{x^{2}}=0$
A) $\{-12,5\}$
B) $\{12,5\}$
C) $\{12,-5\}$
D)...

Question 5

Solve by any method.
-$39 \mathrm{~m}^{2}+96 \mathrm{~m}+55=0$
A) $\left\{\frac{-96+\sqrt{159}}{39}, \frac{-96-\sqrt{159}}{39}\right\}$

B) $\left\{\frac{-48+\sqrt{4449}}{39}, \frac{-48-\sqrt{4449}}{39}\right\}$

C) $\left\{\frac{-48+\sqrt{159}}{39}, \frac{-48-\sqrt{159}}{39}\right\}$

D) $\left\{\frac{-48+\sqrt{159}}{78}, \frac{-48-\sqrt{159}}{78}\right\}$

Accepted Answer

To solve the quadratic equation $39m^2 + 96m + 55 = 0$, we use the quadratic formula $m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 39$, $b = 96$, and $c = 55$. Plugging these values into the formula gives $m = \frac{-96 \pm \sqrt{96^2 - 4(39)(55)}}{2(39)}$. Simplifying inside the square root gives $96^2 - 4(39)(55) = 9216 - 8580 = 636$, which is incorrect, indicating a mistake in my calculation. Let's correct this: the discriminant $D = b^2 - 4ac = 96^2 - 4*39*55$, which actually simplifies to $9216 - 8580 = 636$, and my initial simplification was correct. However, the correct simplification of the square root and the coefficients was not accurately applied to the choices provided. The correct calculation should focus on simplifying the discriminant correctly and applying the quadratic formula accurately. Upon reevaluation, the correct discriminant calculation should be $b^2 - 4ac = 96^2 - 4*39*55 = 9216 - 8580 = 636$, and using this in the quadratic formula gives $m = \frac{-96 \pm \sqrt{636}}{2*39}$. Simplifying $\sqrt{636}$ was not directly relevant to the choices given, indicating a mistake in my explanation process. The correct approach involves directly comparing the provided options with the quadratic formula's structure and the given equation's coefficients.Given the error in calculation and explanation, the focus should be on correctly applying the quadratic formula and accurately identifying the option that matches the calculation results based on the given equation's coefficients. The correct answer should reflect the values of $m$ that satisfy the original equation, considering the proper application of the quadratic formula and the accurate calculation of the discriminant. Reevaluating the correct approach involves accurately applying the quadratic formula with the given coefficients and ensuring the discriminant and the formula's application align with the options provided. The correct answer choice should directly correspond to the results of these calculations, ensuring the values of $m$ are accurately determined based on the quadratic equation provided.

Question 6

Solve by any method.
-$16 \mathrm{x}^{4}-41 \mathrm{x}^{2}+25=0$
A) $\left\{-\frac{5}{4},-1,1, \frac{5}{4}\right\}$
B) $\left\{-1,-\frac{4}{5}\right\}$
C) $\left\{-1,-\frac{4}{5}, \frac{4}{5}, 1\right\}$
D) $\left\{1, \frac{5}{4}\right\}$

Accepted Answer

This equation can be solved by substituting $u = x^2$, turning it into a quadratic equation $16u^2 - 41u + 25 = 0$. Solving this quadratic equation gives $u = \frac{25}{16}$ and $u = 1$. Substituting back $x^2 = u$, we get $x^2 = \frac{25}{16}$ and $x^2 = 1$, leading to the solutions $x = \pm\frac{5}{4}$ and $x = \pm1$.

Question 7

Solve by any method.
-$(-8 \mathrm{p}-5)^{2}=4(-8 \mathrm{p}-5)+12$

A) $\left\{\frac{3}{8}, \frac{11}{8}\right\}$

B) $\left\{-\frac{3}{8},-\frac{11}{8}\right\}$

C) $\{-2,6\}$

D) $\left\{\frac{7}{8},-\frac{1}{8}\right\}$

Accepted Answer

Expanding and simplifying the given equation $(-8p-5)^2 = 4(-8p-5) + 12$ leads to a quadratic equation in terms of $p$. Solving this quadratic equation yields the roots $p = -\frac{3}{8}$ and $p = -\frac{11}{8}$, which matches option B.

Question 8

Solve $S=\frac{1}{2} \pi r^{2}+(r+5) h$ for $r$.

A) $r=\frac{-h \pm \sqrt{h^{2}-10 \pi h+2 \pi S}}{\pi}$

B) $r=\frac{-h \pm \sqrt{h^{2}-10 \pi h-2 \pi S}}{\pi}$

C) $r=\frac{-h \pm \sqrt{h^{2}+10 \pi h+2 \pi S}}{\pi}$

D) $r=\frac{h \pm \sqrt{h^{2}-10 \pi h+2 \pi S}}{\pi}$

Accepted Answer

The answer of Solve $S=\frac{1}{2} \pi r^{2}+(r+5) h$ for $r$.

A)...

Question 9

Ron can mow the lawn in two hours more time than Paul. Working together they can mow the lawn in 4 hours. How long does it take each of them working alone? Round your answers to the nearest tenth of an hour, if necessary.
A) Paul: $6 \mathrm{hr}$
Ron: $10 \mathrm{hr}$

B) Paul: $7.1 \mathrm{hr}$
Ron: $9.1 \mathrm{hr}$

C) Paul: $9.1 \mathrm{hr}$
Ron: $7.1 \mathrm{hr}$

D) Paul: $10 \mathrm{hr}$
Ron: $6 \mathrm{hr}$

Accepted Answer

Let $x$ be the time it takes for Paul to mow the lawn alone. Then, Ron takes $x + 2$ hours. The rate at which Paul works is $1/x$ of the lawn per hour, and Ron's rate is $1/(x + 2)$. Together, their rate is $1/4$ of the lawn per hour since they can complete the task in 4 hours. Setting up the equation: $1/x + 1/(x + 2) = 1/4$. Solving this equation yields $x \approx 7.1$ hours for Paul and $x + 2 \approx 9.1$ hours for Ron.

Question 10

A jet plane traveling at a constant speed goes 1300 miles with the wind, then turns around and travels for 1100 miles against the wind. If the speed of the wind is $50 \mathrm{mph}$ and the total flight took 4 hours, find the speed of the plane in still air.
A) $575 \mathrm{mph}$
B) $600 \mathrm{mph}$
C) $485 \mathrm{mph}$
D) $655 \mathrm{mph}$

Accepted Answer

Let the speed of the plane in still air be $x$ mph. With the wind, the plane's speed is $x + 50$ mph, and against the wind, it's $x - 50$ mph. The total time taken is the sum of the time with the wind and the time against the wind, which equals 4 hours. So, we have the equation: $\frac{1300}{x + 50} + \frac{1100}{x - 50} = 4$. Solving this equation for $x$ gives $x = 600$ mph.

Question 11

A rectangular garden has dimensions of 19 feet by 15 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 552 square feet?
A) $8.5 \mathrm{ft}$
B) $7 \mathrm{ft}$
C) $8 \mathrm{ft}$
D) $6 \mathrm{ft}$

Accepted Answer

The area of the garden is $19 \times 15 = 285$ square feet. The total area, including the path, is $285 + 552 = 837$ square feet. Let the width of the path be $x$. The dimensions of the garden plus the path are $(19 + 2x)$ and $(15 + 2x)$, so the equation is $(19 + 2x)(15 + 2x) = 837$. Solving for $x$, we find $x = 6$ feet.

Question 12

Which one of the following most closely resembles the graph of $f(x)=a(x-h)^{2}+k$ if $a>0, h<0$, and $\mathrm{k}>0$ ?

A)

B)

C)

D)

Accepted Answer

The answer of Which one of the following most closely...

Question 13

Graph the parabola. Identify the vertex, axis, domain and range.
-$f(x)=-\frac{5}{6} x^{2}+6$

A) Vertex: (0, - 6); Axis: $x=0$;
Domain: $(\infty, \infty)$; Range: $[-6, \infty)$

B) Vertex: $\left(0,-\frac{5}{6}\right)$; Axis: $x=0$;
Domain: $(\infty, \infty)$; Range: $\left[-\frac{5}{6}, \infty\right)$

C) Vertex: $(0,6)$; Axis: $x=0$;
Domain: $(\infty, \infty)$; Range: $[6, \infty)$

D) Vertex: $(0,6)$; Axis: $x=0$;
Domain: $(\infty, \infty)$; Range: $[6, \infty)$

Accepted Answer

The answer of Graph the parabola. Identify the vertex, axis,...

Question 14

Graph the parabola. Identify the vertex, axis, domain and range.
-$f(x)=-x^{2}+2 x+2$

A) Vertex: $(-1,1)$; Axis: $\mathrm{x}=-1$;
Domain: $(\infty, \infty)$; Range: $[1, \infty)$

B) Vertex: (1, -1); Axis: $\mathrm{x}=1$;
Domain: $(\infty, \infty)$; Range: $[1, \infty)$

C) Vertex: (1, -1); Axis: $\mathrm{x}=1$;
Domain: $(\infty, \infty)$; Range: $[1, \infty)$

D) Vertex: $(1,3)$; Axis: $\mathrm{x}=1$;
Domain: $(\infty, \infty)$; Range: $(\infty, 3]$

Accepted Answer

The answer of Graph the parabola. Identify the vertex, axis,...

Question 15

Graph the parabola. Identify the vertex, axis, domain and range.
-$x=(y+1)^{2}-6$

A) Vertex: (1, -9); Axis: $\mathrm{x}=-1$;
Domain: $(\infty, \infty)$; Range: $(\infty,-6]$

B) Vertex: $(-6,-1)$; Axis : $y=-1$;
Domain: $(\infty,-6]$; Range: $(\infty, \infty)$

C) Vertex: $(-1,-7)$; Axis: $\mathrm{x}=-1$;
Domain: $(\infty, \infty)$; Range: $[-6, \infty)$
232
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D) Vertex: $(-6,-1)$; Axis : $y=-1$;
Domain: $[-6, \infty)$; Range: $(\infty, \infty)$

Accepted Answer

The answer of Graph the parabola. Identify the vertex, axis,...

Question 16

The percent increase for in-state tuition at a certain public university during the years 1991 through 1999 can be modeled by the quadratic function defined by
$$f(x)=0.161 x^{2}-2.09 x+10.6$$
Where $x=1$ represents $1991, x=2$ represents 1992, and so on.
(i) Based on this model, by what percent (to the nearest tenth) did tuition increase in 1993?
(ii) In what year was the minimum tuition increase? (Round down to the nearest year.) To the nearest tenth, by what percent did tuition increase that year?
A) (i) 25.1%;
(ii) 1997;
33.1%

B) (i) 7.1%;
(ii) 1996;
3)9%
C) (i) 5.8%;
(ii) 1996;
3.9%

D) (i) 4.8%;
(ii) 1995;
4.2%

Accepted Answer

For (i), substitute $x = 3$ into the function to find the percent increase in 1993: $f(3) = 0.161(3)^2 - 2.09(3) + 10.6 = 5.8\%$. For (ii), find the vertex of the parabola, which gives the minimum point. The vertex $x$ value is $-b/(2a) = 6.49$, which rounds down to 1996. Substitute $x = 6$ into the function to find the percent increase: $f(6) = 3.9\%$.

Question 17

The length and width of a rectangular playlot have a sum of 82 yards. What dimensions would give the maximum area?
A) Length 31 yards and width 51 yards
B) Length 40 yards and width 42 yards
C) Length 41 yards and width 41 yards
D) Length 32 yards and width 50 yards

Accepted Answer

The maximum area for a rectangle given a fixed perimeter is achieved when the rectangle is a square. Since the sum of the length and width is 82 yards, dividing this equally gives dimensions of 41 yards by 41 yards, maximizing the area.

Question 18

Solve, and graph the solution set.
-$4 x^{2}+17 x \leq 15$

A) $ (-\infty,-5] \cup\left[\frac{3}{4}, \infty\right] $

B) $ \left[-5, \frac{3}{4}\right] $

C) $ [-5, \infty) $

D) $ \left[-\infty, \frac{3}{4}\right] $

Accepted Answer

The answer of Solve, and graph the solution set.
-\(4 x^{2}+17...

Question 19

Solve, and graph the solution set.
-$\frac{8}{p-3}<1$

A) $ (3,11) $

B) $ (\infty, 3) \cup(11, \infty) $

C) $ [3,11] $

D) $ (\infty, 3] \cup[11, \infty) $

Accepted Answer

The answer of Solve, and graph the solution set.
-$\frac{8}{p-3}<1$

A)...

Quiz 9: Quadratic Equations, Inequalities, and Functions

Solve by using the quadratic formula. - $7 x=\sqrt{\frac{49 x-6}{2}}$

Decide whether the statement is true or false. If it is false, tell why. -If $k$ is a positive number, then the equation $-x^{2}=-100 k$ has no real solutions.

What is the discriminant for $4 x^{2}-12 x+9=0$ ? How many and what type of solutions does this equation have?

Solve by any method. - $1-\frac{7}{x}-\frac{60}{x^{2}}=0$

Solve by any method. - $39 \mathrm{~m}^{2}+96 \mathrm{~m}+55=0$

Solve by any method. - $16 \mathrm{x}^{4}-41 \mathrm{x}^{2}+25=0$

Solve by any method. - $(-8 \mathrm{p}-5) ^{2}=4(-8 \mathrm{p}-5) +12$

Solve $S=\frac{1}{2} \pi r^{2}+(r+5) h$ for $r$ .

Ron can mow the lawn in two hours more time than Paul. Working together they can mow the lawn in 4 hours. How long does it take each of them working alone? Round your answers to the nearest tenth of an hour, if necessary.

A jet plane traveling at a constant speed goes 1300 miles with the wind, then turns around and travels for 1100 miles against the wind. If the speed of the wind is $50 \mathrm{mph}$ and the total flight took 4 hours, find the speed of the plane in still air.

A rectangular garden has dimensions of 19 feet by 15 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 552 square feet?

Which one of the following most closely resembles the graph of $f(x) =a(x-h) ^{2}+k$ if $a>0, h<0$ , and $\mathrm{k}>0$ ?

The length and width of a rectangular playlot have a sum of 82 yards. What dimensions would give the maximum area?

Solve, and graph the solution set. - $4 x^{2}+17 x \leq 15$ $Solve, and graph the solution set. - 4 x^{2}+17 x \leq 15 A) (-\infty,-5] \cup\left[\frac{3}{4}, \infty\right] B) \left[-5, \frac{3}{4}\right] C) [-5, \infty) D) \left[-\infty, \frac{3}{4}\right]$

Solve, and graph the solution set. - $\frac{8}{p-3}<1$ $Solve, and graph the solution set. - \frac{8}{p-3}<1 A) (3,11) B) (\infty, 3) \cup(11, \infty) C) [3,11] D) (\infty, 3] \cup[11, \infty)$

Quiz 9: Quadratic Equations, Inequalities, and Functions

Solve by using the quadratic formula. - 7x=49x−627 x=\sqrt{\frac{49 x-6}{2}}7x=249x−6​​

Decide whether the statement is true or false. If it is false, tell why. -If kkk is a positive number, then the equation −x2=−100k-x^{2}=-100 k−x2=−100k has no real solutions.

What is the discriminant for 4x2−12x+9=04 x^{2}-12 x+9=04x2−12x+9=0 ? How many and what type of solutions does this equation have?

Solve by any method. - 1−7x−60x2=01-\frac{7}{x}-\frac{60}{x^{2}}=01−x7​−x260​=0

Solve by any method. - 39 m2+96 m+55=039 \mathrm{~m}^{2}+96 \mathrm{~m}+55=039 m2+96 m+55=0

Solve by any method. - 16x4−41x2+25=016 \mathrm{x}^{4}-41 \mathrm{x}^{2}+25=016x4−41x2+25=0

Solve by any method. - (−8p−5) 2=4(−8p−5) +12(-8 \mathrm{p}-5) ^{2}=4(-8 \mathrm{p}-5) +12(−8p−5) 2=4(−8p−5) +12

Solve S=12πr2+(r+5) hS=\frac{1}{2} \pi r^{2}+(r+5) hS=21​πr2+(r+5) h for rrr .

Ron can mow the lawn in two hours more time than Paul. Working together they can mow the lawn in 4 hours. How long does it take each of them working alone? Round your answers to the nearest tenth of an hour, if necessary.

A jet plane traveling at a constant speed goes 1300 miles with the wind, then turns around and travels for 1100 miles against the wind. If the speed of the wind is 50mph50 \mathrm{mph}50mph and the total flight took 4 hours, find the speed of the plane in still air.

A rectangular garden has dimensions of 19 feet by 15 feet. A gravel path of equal width is to be built around the garden. How wide can the path be if there is enough gravel for 552 square feet?

Which one of the following most closely resembles the graph of f(x) =a(x−h) 2+kf(x) =a(x-h) ^{2}+kf(x) =a(x−h) 2+k if a>0,h<0a>0, h<0a>0,h<0 , and k>0\mathrm{k}>0k>0 ?

Graph the parabola. Identify the vertex, axis, domain and range. - f(x) =−56x2+6f(x) =-\frac{5}{6} x^{2}+6f(x) =−65​x2+6

Graph the parabola. Identify the vertex, axis, domain and range. - f(x) =−x2+2x+2f(x) =-x^{2}+2 x+2f(x) =−x2+2x+2

Graph the parabola. Identify the vertex, axis, domain and range. - x=(y+1) 2−6x=(y+1) ^{2}-6x=(y+1) 2−6

The length and width of a rectangular playlot have a sum of 82 yards. What dimensions would give the maximum area?

Solve, and graph the solution set. - 4x2+17x≤154 x^{2}+17 x \leq 154x2+17x≤15

Solve, and graph the solution set. - 8p−3<1\frac{8}{p-3}<1p−38​<1

Solve by using the quadratic formula. - $7 x=\sqrt{\frac{49 x-6}{2}}$

Decide whether the statement is true or false. If it is false, tell why. -If $k$ is a positive number, then the equation $-x^{2}=-100 k$ has no real solutions.

What is the discriminant for $4 x^{2}-12 x+9=0$ ? How many and what type of solutions does this equation have?

Solve by any method. - $1-\frac{7}{x}-\frac{60}{x^{2}}=0$

Solve by any method. - $39 \mathrm{~m}^{2}+96 \mathrm{~m}+55=0$

Solve by any method. - $16 \mathrm{x}^{4}-41 \mathrm{x}^{2}+25=0$

Solve by any method. - $(-8 \mathrm{p}-5) ^{2}=4(-8 \mathrm{p}-5) +12$

Solve $S=\frac{1}{2} \pi r^{2}+(r+5) h$ for $r$ .

A jet plane traveling at a constant speed goes 1300 miles with the wind, then turns around and travels for 1100 miles against the wind. If the speed of the wind is $50 \mathrm{mph}$ and the total flight took 4 hours, find the speed of the plane in still air.

Which one of the following most closely resembles the graph of $f(x) =a(x-h) ^{2}+k$ if $a>0, h<0$ , and $\mathrm{k}>0$ ?

Solve, and graph the solution set. - $4 x^{2}+17 x \leq 15$ $Solve, and graph the solution set. - 4 x^{2}+17 x \leq 15 A) (-\infty,-5] \cup\left[\frac{3}{4}, \infty\right] B) \left[-5, \frac{3}{4}\right] C) [-5, \infty) D) \left[-\infty, \frac{3}{4}\right]$

Solve, and graph the solution set. - $\frac{8}{p-3}<1$ $Solve, and graph the solution set. - \frac{8}{p-3}<1 A) (3,11) B) (\infty, 3) \cup(11, \infty) C) [3,11] D) (\infty, 3] \cup[11, \infty)$