Chapter 5

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

What are the solutions of the equation \((2 x-7)^{2}=25 ?\) $$ \begin{array}{lllll}{\text { F. } 6,-6} & {\text { G. } 6,1} & {\text { H. } 6,-1} & {\text { J. }-6,-1}\end{array} $$

Determine \(a\) and \(k\) so both points are on the graph of the function. $$ (1,11),(2,-19) ; y=a(x+1)^{2}+k $$

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

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

Simplify each expression. $$ (4-9 i)(3+8 i) $$

Show that the product of a nonzero complex number \(a+b i\) and its conjugate (as described in Exercise 67 ) is a real number.

Find the sum of the solutions to the equation \(x^{2}+2 x-15=0\) $$ \begin{array}{llll}{\text { A. } 8} & {\text { B. }-8} & {\text { C. } 2} & {\text { D. }-2}\end{array} $$

Determine \(a\) and \(k\) so both points are on the graph of the function. $$ (-2,6),(3,1) ; y=a(x-3)^{2}+k $$

Critical Thinking Explain how to factor \(4 x^{4}+24 x^{3}+32 x^{2}\)

Use the Quadratic Formula to solve each equation for \(x\) in terms of \(a\) $$ 2 a^{2} x^{2}-6 a x=-5 $$

Find the product of the solutions to the equation \(x^{2}-8 x=9\) $$ \begin{array}{llll}{\mathrm{F} .6} & {\text { G. }-6} & {\text { H. } 9} & {\text { J. }-9}\end{array} $$

Determine \(a\) and \(k\) so both points are on the graph of the function. $$ (-2,10),(1,-34) ; y=2 a(x+1)^{2}+k $$

Factor each expression completely. $$ 0.25 t^{2}-0.16 $$

Use the Quadratic Formula to solve each equation for \(x\) in terms of \(a\) $$ 5 a^{2} x^{2}-10 a x=12 $$

Find a quadratic model for each set of data. $$ \left(-1, \frac{1}{2}\right),(0,2),(2,2) $$

Which equation has \(-\frac{2}{5}\) as a solution? $$ \begin{array}{ll}{\text { A. }(2 x-5)(x+1)=0} & {\text { B. }(2 x+5)(x+1)=0} \\\ {\text { C. }(5 x-2)(x+1)=0} & {\text { D. }(5 x+2)(x+1)=0}\end{array} $$

Factor each expression completely. $$ 8100 x^{2}-10,000 $$

Find a quadratic model for each set of data. $$ (0,2),(1,0),(2,4) $$

Use the Quadratic Formula to solve each equation for \(x\) in terms of \(a\) $$ x^{2}+2 a x=25 a^{2} $$

How many times does the graph of \(y=x^{2}-4 x+5\) cross the \(x\) -axis? $$ \begin{array}{lllll}{\text { F. } 0} & {\text { 6. } 1} & {\text { H. } 2} & {\text { 1. } 33}\end{array} $$

Factor each expression completely. $$ 3600 z^{2}-4900 $$

What is the vertex of \(y=-2 x^{2}-4 x-5 ?\) $$ \begin{array}{llll}{\text { A. }(-2,-3)} & {\text { B. }(1,-3)} & {} & {\text { C. }(1,-11)} & {\text { D. }(-1,-3)}\end{array} $$

Use Cramer's Rule to solve each system. $$ \left\\{\begin{array}{l}{2 x+y=4} \\ {3 x-y=6}\end{array}\right. $$

Use the Quadratic Formula to prove each statement. a. The sum of the solutions of the quadratic equation \(a x^{2}+b x+c=0\) is \(-\frac{b}{a}\) b. The product of the solutions of the quadratic equation \(a x^{2}+b x+c=0\) is \(\frac{c}{a}\)

Critical Thinking Graph \(1+3 i\) and \(6+2 i .\) Also graph their sum. Draw the quadrilateral that has three points and the origin as vertices. Repeat with other complex numbers. What do you notice?

The equation \(x^{2}-3 x+a=0\) has two roots. One root of the equation is \(2 .\) What is the other root? $$ \begin{array}{lllll}{\text { A. }-2} & {\text { B. }-1} & {\text { C. } 1} & {\text { D. } 3}\end{array} $$

Open-Ended Write a quadratic function in vertex form for which the graph has a vertex at \((-2,5) .\) Rewrite the function in standard form.

What is the \(y\) -intercept of \(y=(x+1)^{2}-2 ?\) F. \((0,-1)\) G. \((0,-3)\) H. \((0,1)\) J. \((0,-2)\)

Use Cramer's Rule to solve each system. $$ \left\\{\begin{aligned} 2 x+y &=7 \\\\-2 x+5 y &=-1 \end{aligned}\right. $$

What is the discriminant of \(q x^{2}+r x+s=0 ?\) $$ \begin{array}{lllll}{\text { A. }|a+b|} & {\text { B. } q^{2}-4 r s} & {\text { C. } r^{2}-4 q s} & {\text { D. } s^{2}-4 q r}\end{array} $$

What is the number \(\sqrt{-225}+36\) when written in the form \(a+b i ?\) $$\begin{array}{llll}{\text { A. }-15+6 i} & {\text { B. } 6+15 i} & {\text { C. } 6-15 i} & {\text { D. } 36+15 i}\end{array}$$

A quadratic equation has solutions 3 and \(-4 .\) Write a possible equation.

Determine whether the function \(f(x)=0.25(2 x-15)^{2}+150\) has a maximum or a minimum value. Then find the value.

Factor each expression completely. $$ (x-2)^{2}-15(x-2)+56 $$

What is the maximum area in square units of a rectangle with a perimeter of 128 units? $$ \begin{array}{llll}{\text { A. } 4096} & {\text { B. } 1024} & {\text { C. } 256} & {\text { D. } 32}\end{array} $$

How many different real solutions are there for \(2 x^{2}-3 x+5=0 ?\) F. 0 G. 1 H. 2 J. \(i\)

Use Cramer's Rule to solve each system. $$ \left\\{\begin{array}{l}{2 x+4 y=10} \\ {3 x+5 y=14}\end{array}\right. $$

How can you rewrite the expression \((8-5 i)^{2}\) in the form \(a+b i ?\) F. \(39+80 i\) G. \(39-80 i\) H. \(69+80 i\) 1\. \(69-80 i\)

One solution to the equation \(x^{2}+b x-20=0\) is \(5 .\) Find the other solution.

Critical Thinking Describe the differences between the graphs of \(y=(x+6)^{2}\) and \(y=(x-6)^{2}+7\)

Factor each expression completely. $$ 6(x+5)^{2}-5(x+5)+1 $$

The vertex of the graph of \(y=-x^{2}-16 x-62\) lies in which quadrant? \(\mathrm{F} . \mathrm{IV}\) G. III H. II J. 1

Which equation has \(-3 \pm 5 i\) as its solutions? $$ A \cdot x^{2}+6 x=-34 \quad \text { B. } x^{2}+6 x=-14 \quad \text { C. } x^{2}+3 x=4 \quad \text { D. } x^{2}+3 x=2 $$

What are the solutions of \(-4 x^{2}-72=0 ?\) $$\begin{array}{llll}{\text { A. } \pm 2 i \sqrt{3}} & {\text { B. } \pm 3 i \sqrt{2}} & {\text { C. } \pm 2 \sqrt{3}} & {\text { D. } \pm 3 \sqrt{2}}\end{array}$$

Factor each expression. $$ 3 x^{2}-4 x+1 $$

a. In the function \(y=a x^{2}+b x+c, c\) represents the \(y\) -intercept. Find the value of the \(y\) -intercept in the function \(y=a(x-h)^{2}+k\) b. Under what conditions does \(k\) represent the \(y\) -intercept?

What percent of nonzero integers have squares that are odd numbers? $$ \begin{array}{lllll}{\text { A. } 25} & {\text { B. } 50} & {\text { C. } 75} & {\text { D. } 100}\end{array} $$

What is the discriminant of a quadratic equation, and what does its value tell you about the solution(s) of the equation?

Which description of the graph of \(y=a x^{2}+b x+c\) is NOT possible? F. There are two \(x\) -intercepts, the vertex is below the \(x\) -axis, and \(a>0\) . G. There is one \(x\) -intercept and the vertex is on the \(x\) -axis. H. There are two \(x\) -intercepts, the vertex is below the \(x\) -axis, and \(a<0\) . J. There are no \(x\) -intercepts, the vertex is above the \(x\) -axis, and \(a>0\) .