Question 1

Solve.
-$$\sqrt [ 3 ] { 4 x + 7 } = - 3$$

Accepted Answer

To solve the equation $\sqrt[3]{4x + 7} = -3$, cube both sides to get rid of the cube root: $(\sqrt[3]{4x + 7})^3 = (-3)^3$, which simplifies to $4x + 7 = -27$. Solving for $x$, subtract 7 from both sides to get $4x = -34$, and then divide by 4 to find $x = -\frac{34}{4} = -\frac{17}{2}$.

Question 2

Solve graphically. Round solutions to three decimal places, where appropriate.
-$$3 x ^ { 2 } - 5 = 6 x$$

Accepted Answer

The answer of Solve graphically. Round solutions to three decimal...

Question 3

Solve.
-$\sqrt { 3 x + 1 } = 3 + \sqrt { x - 4 }$

Accepted Answer

The answer of Solve.
-\(\sqrt { 3 x + 1 }...

Question 4

Solve.
-$$\frac { x } { x - 5 } - \frac { 5 } { x + 5 } = \frac { 50 } { x ^ { 2 } - 25 }$$

Accepted Answer

The equation involves a denominator that can become zero if $x = \pm 5$, which are the roots of $x^2 - 25$. Since dividing by zero is undefined, $x$ cannot be $\pm 5$, leading to no solution.

Question 5

Simplify.
-$( - i ) ^ { 42 }$

Accepted Answer

The answer of Simplify.
-\(( - i ) ^ { 42...

Question 6

Use the quadratic formula to find the exact solutions.
-$$3 x ^ { 2 } - 10 x - 8 = 0$$

Accepted Answer

The answer of Use the quadratic formula to find the...

Question 7

Simplify. Write your answer in the form a + bi, where a and b are real numbers.
-$$( - 6 - 4 i ) - ( 5 - i )$$

Accepted Answer

The answer of Simplify. Write your answer in the form...

Question 8

Express the number in terms of i.
-$- \sqrt { - 19 }$

Accepted Answer

The answer of Express the number in terms of i.
-\(-...

Question 9

Simplify. Write your answers in the form of a+bi, where a and b are real numbers.
-$$( 7 - 3 \mathrm { i } ) ^ { 2 }$$

Accepted Answer

To simplify $(7 - 3i)^2$, we first expand the expression: $(7 - 3i)(7 - 3i) = 49 - 21i - 21i + 9i^2$. Since $i^2 = -1$, we get $49 - 42i - 9 = 40 - 42i$.

Question 10

Use the quadratic formula to find the exact solutions.
-$$3 x ^ { 2 } - 20 x - 7 = 0$$

Accepted Answer

The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ for a quadratic equation $ax^2 + bx + c = 0$. Here, $a = 3$, $b = -20$, and $c = -7$. Plugging these into the formula gives $$x = \frac{20 \pm \sqrt{(-20)^2 - 4(3)(-7)}}{2(3)} = \frac{20 \pm \sqrt{400 + 84}}{6} = \frac{20 \pm \sqrt{484}}{6} = \frac{20 \pm 22}{6}$$, which simplifies to $x = -\frac{1}{3}$ or $x = 7$.

Question 11

Find the range of the given function.
-$$f ( x ) = - 2 x ^ { 2 } - 12 x - 24$$

Accepted Answer

The function is a downward-opening parabola because the coefficient of $x^2$ is negative. To find the vertex, we use the formula $x = -\frac{b}{2a}$, where $a = -2$ and $b = -12$. This gives $x = -\frac{-12}{-4} = 3$. Plugging $x = 3$ into the function gives the y-coordinate of the vertex as $f(3) = -2(3)^2 - 12(3) - 24 = -18 - 36 - 24 = -78$. However, the calculation for the y-coordinate of the vertex seems to have been done incorrectly. Correctly calculating the y-coordinate of the vertex: $f(3) = -2(3)^2 - 12(3) - 24 = -18 - 36 - 24 = -78$ is incorrect. The correct calculation should be: $f(3) = -2(9) - 12(3) - 24 = -18 - 36 - 24 = -78$, which is indeed the process, but the interpretation of the result was mistaken in the explanation. The correct y-coordinate after correctly calculating should be found, and it will represent the maximum value of the function since the parabola opens downwards. The correct maximum value (y-coordinate of the vertex) actually is $f(3) = -2(9) - 36 - 24 = -18 - 36 - 24 = -78$, indicating the peak of the parabola. Thus, the range of the function is all real numbers less than or equal to this maximum value. Therefore, the correct range should be identified based on the vertex's correct y-value, leading to the correct option that matches the parabola's characteristics.

Question 12

Solve.
-The number of mosquitoes $\mathrm { M } ( \mathrm { x } )$, in millions, in a certain area depends on the June rainfall $x$, in inches, according to the function $\mathrm { M } ( \mathrm { x } ) = 3 \mathrm { x } - \mathrm { x } ^ { 2 }$. What rainfall produces the maximum number of mosquitoes?

Accepted Answer

To find the maximum number of mosquitoes, we need to find the vertex of the parabola represented by the function $M(x) = 3x - x^2$. The x-coordinate of the vertex can be found using the formula $-\frac{b}{2a}$, where $a = -1$ and $b = 3$. Thus, $-\frac{3}{2(-1)} = 1.5$, indicating that the maximum number of mosquitoes occurs with 1.5 inches of rainfall.

Question 13

Solve.
-$\sqrt { 3 x - 2 } + \sqrt { 11 + x } = - 1$

Accepted Answer

The equation involves square roots, and square roots can only yield non-negative results. The sum of two non-negative numbers cannot be negative, so there is no solution.

Question 14

Simplify. Write your answers in the form of a+bi, where a and b are real numbers.
-$$\frac { 2 - 7 i } { 1 + 6 i } + \frac { 3 - 5 i } { 1 + 6 i }$$

Accepted Answer

The answer of Simplify. Write your answers in the form...

Question 15

Use the given graph to find the x-intercepts and zeros of the function.
-

Accepted Answer

The answer of Use the given graph to find the...

Question 16

Solve the problem.
-A ladder is resting against a wall. The top of the ladder touches the wall at a height of 12 ft. Find the length of the ladder if the length is 4 ft more than its distance from the wall.

Accepted Answer

Using the Pythagorean theorem, let x be the distance from the wall. The ladder's length is x + 4. The equation is x^2 + 12^2 = (x + 4)^2. Solving gives x = 16, so the ladder's length is 20 ft.

Question 17

Find the range of the given function.
-$$f ( x ) = 4 x ^ { 2 } - 40 x + 98$$

Accepted Answer

The range of the function is the set of all possible output values. To find the range, we need to find the minimum and maximum values of the function. The minimum value of the function occurs at the vertex, which is at $x = 5$. The maximum value of the function is infinity, since the function increases without bound as $x$ approaches infinity. Therefore, the range of the function is $[ - 2 , \infty )$.

Question 18

Find the range of the given function.
-$$f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 8 x - \frac { 5 } { 2 }$$

Accepted Answer

The function is a quadratic with a positive leading coefficient, so it opens upwards. The vertex form reveals the minimum value, which is $-\frac{69}{2}$, hence the range is $\left[ - \frac { 69 } { 2 } , \infty \right)$.

Question 19

Use the graph to find the vertex, the axis of symmetry, and the maximum or minimum value of the function.
-

Accepted Answer

The answer of Use the graph to find the vertex,...

Question 20

Simplify.
-¡16

Accepted Answer

The answer of Simplify.
-¡16
...

Quiz 3: Quadratic Functions and Equations; Inequalities

Solve. - $\sqrt [ 3 ] { 4 x + 7 } = - 3$

Solve graphically. Round solutions to three decimal places, where appropriate. - $3 x ^ { 2 } - 5 = 6 x$

Solve. - $\sqrt { 3 x + 1 } = 3 + \sqrt { x - 4 }$

Solve. - $\frac { x } { x - 5 } - \frac { 5 } { x + 5 } = \frac { 50 } { x ^ { 2 } - 25 }$

Simplify. - $( - i ) ^ { 42 }$

Use the quadratic formula to find the exact solutions. - $3 x ^ { 2 } - 10 x - 8 = 0$

Simplify. Write your answer in the form a + bi, where a and b are real numbers. - $( - 6 - 4 i ) - ( 5 - i )$

Express the number in terms of i. - $- \sqrt { - 19 }$

Simplify. Write your answers in the form of a+bi, where a and b are real numbers. - $( 7 - 3 \mathrm { i } ) ^ { 2 }$

Use the quadratic formula to find the exact solutions. - $3 x ^ { 2 } - 20 x - 7 = 0$

Find the range of the given function. - $f ( x ) = - 2 x ^ { 2 } - 12 x - 24$

Solve. - $\sqrt { 3 x - 2 } + \sqrt { 11 + x } = - 1$

Simplify. Write your answers in the form of a+bi, where a and b are real numbers. - $\frac { 2 - 7 i } { 1 + 6 i } + \frac { 3 - 5 i } { 1 + 6 i }$

Use the given graph to find the x-intercepts and zeros of the function. -

Solve the problem. -A ladder is resting against a wall. The top of the ladder touches the wall at a height of 12 ft. Find the length of the ladder if the length is 4 ft more than its distance from the wall.

Find the range of the given function. - $f ( x ) = 4 x ^ { 2 } - 40 x + 98$

Find the range of the given function. - $f ( x ) = \frac { 1 } { 2 } x ^ { 2 } - 8 x - \frac { 5 } { 2 }$

Use the graph to find the vertex, the axis of symmetry, and the maximum or minimum value of the function. -

Simplify. -¡16

Quiz 3: Quadratic Functions and Equations; Inequalities

Solve. - 4x+73=−3\sqrt [ 3 ] { 4 x + 7 } = - 334x+7​=−3

Solve graphically. Round solutions to three decimal places, where appropriate. - 3x2−5=6x3 x ^ { 2 } - 5 = 6 x3x2−5=6x

Solve. - 3x+1=3+x−4\sqrt { 3 x + 1 } = 3 + \sqrt { x - 4 }3x+1​=3+x−4​

Solve. - xx−5−5x+5=50x2−25\frac { x } { x - 5 } - \frac { 5 } { x + 5 } = \frac { 50 } { x ^ { 2 } - 25 }x−5x​−x+55​=x2−2550​

Simplify. - (−i) 42( - i ) ^ { 42 }(−i) 42

Use the quadratic formula to find the exact solutions. - 3x2−10x−8=03 x ^ { 2 } - 10 x - 8 = 03x2−10x−8=0

Simplify. Write your answer in the form a + bi, where a and b are real numbers. - (−6−4i) −(5−i) ( - 6 - 4 i ) - ( 5 - i ) (−6−4i) −(5−i)

Express the number in terms of i. - −−19- \sqrt { - 19 }−−19​

Simplify. Write your answers in the form of a+bi, where a and b are real numbers. - (7−3i) 2( 7 - 3 \mathrm { i } ) ^ { 2 }(7−3i) 2

Use the quadratic formula to find the exact solutions. - 3x2−20x−7=03 x ^ { 2 } - 20 x - 7 = 03x2−20x−7=0

Find the range of the given function. - f(x) =−2x2−12x−24f ( x ) = - 2 x ^ { 2 } - 12 x - 24f(x) =−2x2−12x−24

Solve. - 3x−2+11+x=−1\sqrt { 3 x - 2 } + \sqrt { 11 + x } = - 13x−2​+11+x​=−1

Simplify. Write your answers in the form of a+bi, where a and b are real numbers. - 2−7i1+6i+3−5i1+6i\frac { 2 - 7 i } { 1 + 6 i } + \frac { 3 - 5 i } { 1 + 6 i }1+6i2−7i​+1+6i3−5i​