Chapter 2

For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring. $$ \left(x^{2}-1\right)^{2}+\left(x^{2}-1\right)-12=0 $$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution. $$ x^{2}+x=4 $$

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ i^{8} $$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve for \(h: A=\frac{1}{2} h\left(b_{1}+b_{2}\right)\)

For the following exercises, graph the pair of equations on the same axes, and state whether they are parallel, perpendicular, or neither. $$ \begin{array}{l} x=4 \\ y=-3 \end{array} $$

For each of the following exercises, find and plot the \(x\) - and \(y\) -intercepts, and graph the straight line based on those two points. $$ x-2 y=8 $$

For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring. $$ (x+1)^{2}-8(x+1)-9=0 $$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution. $$ 2 x^{2}-8 x-5=0 $$

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ i^{15} $$

For the following exercises, find the slope of the line that passes through the given points. (5,4) and (7,9)

For each of the following exercises, find and plot the \(x\) - and \(y\) -intercepts, and graph the straight line based on those two points. $$ y-5=5 x $$

For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being \(y 2\) ) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the \(y\) -values of the lines. $$ \frac{1}{2} x+1>\frac{1}{2} x-5 $$

For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring. $$ (x-3)^{2}-4=0 $$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution. $$ 3 x^{2}-5 x+1=0 $$

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ i^{22} $$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Find the dimensions of an American football field. The length is \(200 \mathrm{ft}\) more than the width, and the perimeter is \(1,040 \mathrm{ft}\). Find the length and width. Use the perimeter formula \(P=2 L+2 W\).

For the following exercises, find the slope of the line that passes through the given points. (-3,2) and (4,-7)

For each of the following exercises, find and plot the \(x\) - and \(y\) -intercepts, and graph the straight line based on those two points. $$ 3 y=-2 x+6 $$

For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being \(y 2\) ) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the \(y\) -values of the lines. $$ 4 x+1<\frac{1}{2} x+3 $$

For the following exercises, solve for the unknown variable. $$ x^{-2}-x^{-1}-12=0 $$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution. $$ x^{2}+4 x+2=0 $$

For the following exercises, use a calculator to help answer the questions. Evaluate \((1+i)^{k}\) for \(k=4,8\) and 12 . Predict the value if \(k=16\).

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Distance equals rate times time, \(d=r t\). Find the distance Tom travels if he is moving at a rate of \(55 \mathrm{mi} / \mathrm{h}\) for \(3.5 \mathrm{~h}\).

For the following exercises, find the slope of the line that passes through the given points. (-5,4) and (2,4)

For each of the following exercises, find and plot the \(x\) - and \(y\) -intercepts, and graph the straight line based on those two points. $$ y=\frac{x-3}{2} $$

For the following exercises, write the set in interval notation. $$ \\{x \mid-1

For the following exercises, solve for the unknown variable. $$ \sqrt{|x|^{2}}=x $$

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution. $$ 4+\frac{1}{x}-\frac{1}{x^{2}}=0 $$

For the following exercises, use a calculator to help answer the questions. Evaluate \((1-i)^{k}\) for \(k=2,6\) and \(10 .\) Predict the value if \(k=14\).

For the following exercises, find the slope of the line that passes through the given points. (-1,-2) and (3,4)

For the following exercises, write the set in interval notation. $$ \\{x \mid x \geq 7\\} $$

For the following exercises, solve for the unknown variable. $$ t^{10}-2 t^{5}+1=0 $$

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the \(x\) -intercepts) by using \(2^{\text {nd }}\) CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth. \(\mathrm{Y}_{1}=4 x^{2}+3 x-2\)

For the following exercises, use a calculator to help answer the questions. Evaluate \((1+i)^{k}-(1-i)^{k}\) for \(k=4,8,\) and \(12 .\) Predict the value for \(k=16\).

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. What is the total distance that two people travel in \(3 \mathrm{~h}\) if one of them is riding a bike at \(15 \mathrm{mi} / \mathrm{h}\) and the other is walking at \(3 \mathrm{mi} /\) h?

For the following exercises, find the slope of the line that passes through the given points. (3,-2) and (3,-2)

For the following exercises, write the set in interval notation. $$ \\{x \mid x<4\\} $$

For the following exercises, solve for the unknown variable. $$ \left|x^{2}+2 x-36\right|=12 $$

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the \(x\) -intercepts) by using \(2^{\text {nd }}\) CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth. \(Y_{1}=-3 x^{2}+8 x-1\)

For the following exercises, use a calculator to help answer the questions. Show that a solution of \(x^{6}+1=0\) is \(\frac{\sqrt{3}}{2}+\frac{1}{2} i\).

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. If the area model for a triangle is \(A=\frac{1}{2} b h,\) find the area of a triangle with a height of 16 in. and a base of 11 in.

For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular. \(\begin{array}{ll}(-1,3) & \text { and }(5,1) \\ (-2,3) & \text { and }(0,9)\end{array}\)

For each of the following exercises, use the graph in the figure below. Find the distance that (-3,4) is from the origin.

For the following exercises, use the model for the period of a pendulum, \(T\), such that \(T=2 \pi \sqrt{\frac{L}{g}}\), where the length of the pendulum is \(L\) and the acceleration due to gravity is g. If the acceleration due to gravity is \(9.8 \mathrm{~m} / \mathrm{s}^{2}\) and the period equals \(1 \mathrm{~s}\), find the length to the nearest \(\mathrm{cm}(100 \mathrm{~cm}=1 \mathrm{~m})\).

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the \(x\) -intercepts) by using \(2^{\text {nd }}\) CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth. \(\mathrm{Y}_{1}=0.5 x^{2}+x-7\)

For the following exercises, use a calculator to help answer the questions. Show that a solution of \(x^{8}-1=0\) is \(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\).

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve for \(h: A=\frac{1}{2} b h\)

For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular. $$\begin{array}{l} (2,5) \text { and }(5,9) \\ (-1,-1) \text { and }(2,3) \end{array}$$

For each of the following exercises, use the graph in the figure below. Find the distance that (5,2) is from the origin. Round to three decimal places.

For the following exercises, write the interval in set-builder notation. \((-\infty, 6)\)