Chapter 5

Solve each equation by completing the square. $$ x^{2}+10 x+17=0 $$

Simplify. $$ \frac{5-i \sqrt{3}}{5+i \sqrt{3}} $$

Factor completely. $$ 3 x^{2}+8 x+4 $$

Name the property illustrated by each equation. \(3(6 x-7 y)=3(6 x)+3(-7 y)\)

OPEN ENDED. Give an example of a quadratic function that has a domain of all real numbers and a range of all real numbers less than a maximum value. State the maximum value and sketch the graph of the function.

Solve each equation by completing the square. $$ x^{2}-6 x+18=0 $$

Simplify. $$ \frac{1-i \sqrt{2}}{1+i \sqrt{2}} $$

Factor completely. $$ 6 x^{2}-14 x-12 $$

Name the property illustrated by each equation. \((3+4)+x=3+(4+x)\)

CHALLENGE Write an expression for the minimum value of a function of the form \(y=a x^{2}+c,\) where \(a>0 .\) Explain your reasoning. Then use this function to find the minimum value of \(y=8.6 x^{2}-12.5 .\)

Desiree works in a photography studio and makes a commission of \(\$ 8\) per photo package she sels. On Tuesday, she sold 3 more packages than she sold on Monday. For the two days, Victoria earned \(\$ 264 .\) How many photo packages did she sell on these two days?

Solve each equation by completing the square. $$ 4 x^{2}+8 x=9 $$

Solve each equation, and locate the complex solutions in the complex plane. $$ -3 x^{2}-9=0 $$

The two zeros of a quadratic function are labeled \(x_{1}\) and \(x_{2}\) on the graph. Which expression has the greatest value? A. 2\(x_{1}\) B. \(x_{2}\) C. \(x_{2}-x_{1}\) D. \(x_{2}+x_{1}\)

Name the property illustrated by each equation. \((5 x)(-3 y)(6)=(-3 y)(6)(5 x)\)

State whether each trinomial is a perfect square. If so, factor it. \(x^{2}-5 x-10\)

Determine whether the given value satisfies the inequality. $$ -2 x^{2}+3 < 0 ; x=5 $$

Solve each equation, and locate the complex solutions in the complex plane. $$ -2 x^{2}-80=0 $$

ACT/SAT The graph of which of the following equations is symmetrical about the \(y\) -axis? A \(y=x^{2}+3 x-1\) B \(y=-x^{2}+x\) C \(y=6 x^{2}+9\) D \(y=3 x^{2}-3 x+1\)

State whether each trinomial is a perfect square. If so, factor it. \(x^{2}-14 x+49\)

LAW ENFORCEMENT A certain laser device measures vehicle speed to within 3 miles per hour. If a vehicle's actual speed is 65 miles per hour, write and solve an absolute value equation to describe the range of speeds that might register on this device.

Determine whether the given value satisfies the inequality. $$ 4 x^{2}+2 x-3 \geq 0 ; x=-1 $$

Solve each equation, and locate the complex solutions in the complex plane. $$ \frac{2}{3} x^{2}+30=0 $$

Simplify. \(i^{14}\)

REVIEW In which equation does every real number \(x\) correspond to a nonnegative real number \(y\) ? \(\begin{array}{rl}{\mathbf{F}} & {y=-x^{2}} \\ {\mathbf{G}} & {y=-x} \\\ {\mathbf{H}} & {y=x} \\ {\mathbf{J}} & {y=x^{2}}\end{array}\)

State whether each trinomial is a perfect square. If so, factor it. \(4 x^{2}+12 x+9\)

Determine whether the given value satisfies the inequality. $$ 4 x^{2}-4 x+1 \leq 10 ; x=2 $$

Solve each equation, and locate the complex solutions in the complex plane. $$ \frac{4}{5} x^{2}+1=0 $$

Simplify. \((4-3 i)-(5-6 i)\)

Solve each system of equations by using inverse matrices. $$ \begin{array}{l}{2 x+3 y=8} \\ {x-2 y=-3}\end{array} $$

State whether each trinomial is a perfect square. If so, factor it. \(25 x^{2}+20 x+4\)

Determine whether the given value satisfies the inequality. $$ 6 x^{2}+3 x > 8 ; x=0 $$

Find the values of \(m\) and \(n\) that make each equation true. $$ (m+2 n)+(2 m-n) i=5+5 i $$

Simplify. \((7+2 i)(1-i)\)

Solve each system of equations by using inverse matrices. $$ \begin{array}{l}{x+4 y=9} \\ {3 x+2 y=-3}\end{array} $$

State whether each trinomial is a perfect square. If so, factor it. \(9 x^{2}-12 x+16\)

Find the values of \(m\) and \(n\) that make each equation true. $$ (2 m-3 n) i+(m+4 n)=13+7 i $$

Solve each equation by factoring. \(4 x^{2}+8 x=0\)

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rr}{2} & {5} \\ {-1} & {-2}\end{array}\right] $$

State whether each trinomial is a perfect square. If so, factor it. \(36 x^{2}-60 x+25\)

ELECTRICITY. The impedance in one part of a series circuit is \(3+4 j\) ohms, and the impedance in another part of the circuit is \(2-6 j .\) Add these complex numbers to find the total impedance in the circuit.

Solve each equation by factoring. \(x^{2}-5 x=14\)

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{ll}{4} & {3} \\ {1} & {1}\end{array}\right] $$

OPEN ENDED Write two complex numbers with a product of \(10 .\)

Solve each equation by factoring. \(3 x^{2}+10=17 x\)

Perform the indicated operation, if possible. $$ \left[\begin{array}{rr}{2} & {-1} \\ {0} & {5}\end{array}\right] \cdot\left[\begin{array}{rr}{-3} & {2} \\ {1} & {4}\end{array}\right] $$

Solve each system of equations by using inverse matrices. \(5 x+3 y=-5\) \(7 x+5 y=-11\)

Perform the indicated operation, if possible. $$ \left[\begin{array}{ll}{1} & {-3}\end{array}\right] \cdot\left[\begin{array}{rrr}{4} & {-2} & {1} \\ {-3} & {2} & {0}\end{array}\right] $$

Which One Doesn't Belong? Identify the expression that does not belong with the other three. Explain your reasoning. $$ (3 i)^{2} \quad(2 i)(3 i)(4 i) \quad(6+2 i)-(4+2 i) $$

Solve each system of equations by using inverse matrices. \(6 x+5 y=8\) \(3 x-y=7\)