Chapter 5

Simplify each expression. $$ (10+\sqrt{-9})-(2+\sqrt{-25}) $$

Open-Ended Write a quadratic equation with the given solutions. \(-1\) and \(-6\)

Factor each expression completely. $$ 4 n^{2}-20 n+24 $$

Without graphing, tell how many \(x\) -intercepts each function has. $$ y=3 x^{2}-10 x+6 $$

Solve for \(x\) in terms of \(a\). $$ 3 x^{2}+a x^{2}=9 x+9 a $$

Simplify each expression. $$ (8-\sqrt{-1})-(-3+\sqrt{-16}) $$

Open-Ended Write a quadratic equation with the given solutions. \(\frac{1}{2}\) and \(\frac{2}{3}\)

Factor each expression completely. $$ 3 y^{2}+24 y+45 $$

Without graphing, tell how many \(x\) -intercepts each function has. $$ y=10 x^{2}+13 x-3 $$

Solve for \(x\) in terms of \(a\). $$ 6 a^{2} x^{2}-11 a x=10 $$

Simplify each expression. $$ 2 i(5-3 i) $$

Matrices Find the possible values of \(x\) and \(y .\) (A matrix with exponent 2 means that you multiply the matrix by itself.) $$ \left[\begin{array}{ll}{x} & {2} \\ {3} & {y}\end{array}\right]^{2}=\left[\begin{array}{ll}{22} & {10} \\ {15} & {\mathbb{1}}\end{array}\right] $$

Determine whether each function is written in vertex form. If a function is not in vertex form, rewrite the function. $$ y=-2(x+1)^{2}-1 $$

Factor each expression completely. $$ -x^{2}+5 x-4 $$

Simplify each expression. $$ -5(1+2 i)+3 i(3-4 i) $$

Determine whether each function is written in vertex form. If a function is not in vertex form, rewrite the function. $$ y=x^{2}+2 x+8 $$

Factor each expression completely. $$ 4 x^{2}-22 x+10 $$

What can you add to \(x^{2}+5 x\) to get a perfect square trinomial? \(\begin{array}{llll}{\text { A. } \frac{25}{4}} & {\text { B. } \frac{25}{2}} & {\text { C. } 25} & {\text { D. } 2.5 x}\end{array}\)

Simplify each expression. $$ (3+\sqrt{-4})(4+\sqrt{-1}) $$

Determine whether each function is written in vertex form. If a function is not in vertex form, rewrite the function. $$ y=\frac{3}{10} x^{2}-1 $$

Factor each expression completely. $$ \frac{1}{2} x^{2}-\frac{1}{2} $$

Error Analysis After analyzing a quadratic equation with real coefficients, a student says that the equation has exactly one imaginary solution. Explain how you know that the student is wrong.

Simplify each expression. $$ (-2+\sqrt{-9})(6+\sqrt{-25}) $$

Physics Suppose you throw a ball straight up from the ground with a velocity of 80 \(\mathrm{ft} / \mathrm{s}\) . As the ball moves upward, gravity slows it. Eventually the ball begins to fall back to the ground. The height \(h\) of the ball after \(t\) seconds in the air is given by the quadratic function \(h(t)=-16 t^{2}+80 t .\) a. How high does the ball go? b. For how many seconds is the ball in the air before it hits the ground?

vertex form, rewrite the function. $$ y=-4 x^{2}+6 x+3 $$

Factor each expression completely. $$ -6 z^{2}-600 $$

Use the discriminant to match each function with its graph. $$ f(x)=x^{2}-4 x+2 \quad \text { b. } f(x)=x^{2}-4 x+4 \quad \text { c. } f(x)=x^{2}-4 x+6 $$

What are the solutions of the equation \(x^{2}+10 x+40=5 ?\) \(\begin{array}{llll}{\text { A. } 10 \pm} & {\text { i\sqrt } 5} & {\text { B. } 5 \pm i \sqrt{10}} & {\text { C. }-5 \pm i \sqrt{10}} & {\text { D. }-10 \pm i \sqrt{5}}\end{array}\)

Simplify each expression. $$ (1-\sqrt{-4})(-3-\sqrt{-25}) $$

a. Let \(a>0 .\) Use algebraic or arithmetic ideas to explain why the lowest point on the graph of \(y=a(x-h)^{2}+k\) must occur when \(x=h\) . b. Suppose that the function in part (a) is \(y=a(x-h)^{3}+k .\) Is your reasoning still valid? Explain.

Determine whether each function is written in vertex form. If a function is not in vertex form, rewrite the function. $$ y=0.5 x^{2}+10 $$

A student says that that the graph of \(y=a x^{2}+b x+c\) gets wider as \(a\) increases. a. Error Analysis Use examples to show that the student is wrong. b. Writing Summarize the relationship between \(|a|\) and the width of the graph of \(y=a x^{2}+b x+c\)

Solve \(14 x=x^{2}+36 .\) Show your work.

What are the values of \(x\) that satisfy the equation \(3-27 x^{2}=0 ?\) $$ \begin{array}{ll}{\text { A. } x=\pm 3} & {\text { B. } x=\pm \frac{1}{3}} \\\ {\text { C. } x=\frac{1}{9} \text { or } x=-\frac{1}{9}} & {\text { D. } x=2 \sqrt{6} \text { or } x=-2 \sqrt{6}}\end{array} $$

Agriculture The area in square feet of a rectangular field is \(x^{2}-120 x+3500 .\) The width in feet is \(x-50\) . Find the length.

For each function, the vertex of the function's graph is given. Find \(a\) and \(b\) \ $$ y=a x^{2}+b x-27 ;(2,-3) $$

List the steps for solving the equation \(3 x^{2}-6=-7 x\) by the method of completing the square. Explain each step.

Two complex numbers \(a+b i\) and \(c+d i\) are equal when \(a=c\) and \(b=d .\) Solve each equation for \(x\) and \(y .\) $$ 2 x+3 y i=-14+9 i $$

What are the solutions of the equation \(6 x^{2}+9 x-15=0 ?\) $$ \begin{array}{ll}{\text { F. } 1,-15} & {\text { G. } 1,-\frac{5}{2}} \\\ {\text { H. }-1,-5} & {\text { J. } 3, \frac{5}{2}}\end{array} $$

Determine \(a\) and \(k\) so both points are on the graph of the function. $$ (0,1),(2,1) ; y=a(x-1)^{2}+k $$

Writing Explain how to factor \(3 x^{2}+6 x-72\) completely.

For each function, the vertex of the function's graph is given. Find \(a\) and \(b\) \ $$ y=a x^{2}+b x+5 ;(-1,4) $$

Write a quadratic equation with the given solutions. $$ \frac{3+\sqrt{5}}{2}, \frac{3-\sqrt{5}}{2} $$

Simplify each expression. $$ (2-3 i)+(-4+5 i) $$

Two complex numbers \(a+b i\) and \(c+d i\) are equal when \(a=c\) and \(b=d .\) Solve each equation for \(x\) and \(y .\) $$ 3 x+19 i=16-8 y i $$

For which equation is \(-3\) NOT a solution? $$ \begin{array}{ll}{\text { A. } x^{2}-2 x-15=0} & {\text { B. } x^{2}-21=4 x} \\\ {\text { C. } 2 x^{2}+12 x=-18} & {\text { D. } 9+x^{2}=0}\end{array} $$

Determine \(a\) and \(k\) so both points are on the graph of the function. $$ (-3,2),(0,11) ; y=a(x+2)^{2}+k $$

For each function, the vertex of the function's graph is given. Find \(a\) and \(b\) \ $$ y=a x^{2}+b x+8 ;(2,-4) $$

Write a quadratic equation with the given solutions. $$ \frac{-5+\sqrt{13}}{2}, \frac{-5-\sqrt{13}}{2} $$

Simplify each expression. $$ (7+3 i)-(2+i) $$