Chapter 5

In \(19-25,\) express each answer in simplest radical form. Check each answer. The length of a rectangle is 2 feet more than the width. The area of the rectangle is 2 square feet. What are the dimensions of the rectangle?

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ x^{2}-8 x+4=0 $$

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=x^{2}-8 x+6} \\ {2 x-y=10}\end{array} $$

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ \mathrm{f}(x)=5 x^{2}+4 x+1 \text { and } a=-1 $$

In \(15-23 :\) a. Find the value of the discriminant and determine if the roots of the quadratic equation are \((1)\) rational and unequal, \((2)\) rational and equal, \((3)\) irrational and unequal, or \((4)\) not real numbers. b. Use any method to find the real roots of the equation if they exist. $$ 4 x-1=4 x^{2} $$

In \(18-25,\) write the complex conjugate of each number. $$ -6-9 i $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{-144}+\sqrt{-1} $$

In \(19-25,\) express each answer in simplest radical form. Check each answer. The length of a rectangle is 4 centimeters more than the width. The measure of a diagonal is 10 centimeters. Find the dimensions of the rectangle.

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ 2 x^{2}+12 x+3=0 $$

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{2 x^{2}-3 x+3-y=0} \\ {y-2 x=1}\end{array} $$

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ \mathrm{f}(x)=-x^{3}+x-24 \text { and } a=-3 $$

In \(15-23 :\) a. Find the value of the discriminant and determine if the roots of the quadratic equation are \((1)\) rational and unequal, \((2)\) rational and equal, \((3)\) irrational and unequal, or \((4)\) not real numbers. b. Use any method to find the real roots of the equation if they exist. $$ 2 x^{2}-3 x-5=0 $$

In \(18-25,\) write the complex conjugate of each number. $$ \frac{1}{2}-3 i $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{-49}-\sqrt{-16} $$

The length of the base of a triangle is 6 feet more than the length of the altitude to the base. The area of the triangle is 18 square feet. Find the length of the base and of the altitude to the base.

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ 2 x^{2}-6 x+3=0 $$

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=x^{2}-4 x+5} \\ {2 y=x+6}\end{array} $$

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ \mathrm{f}(x)=x^{4}+x^{2}+x+1 \text { and } a=0 $$

One of the roots is given. Find the other root. \(6 x^{2}-x+c=0 ;-\frac{2}{3}\)

In \(15-23 :\) a. Find the value of the discriminant and determine if the roots of the quadratic equation are \((1)\) rational and unequal, \((2)\) rational and equal, \((3)\) irrational and unequal, or \((4)\) not real numbers. b. Use any method to find the real roots of the equation if they exist. $$ 3 x-x^{2}=5 $$

In \(18-25,\) write the complex conjugate of each number. $$ -4+\frac{1}{3} i $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{-12}+\sqrt{-27}-\sqrt{-75} $$

In \(19-25,\) express each answer in simplest radical form. Check each answer. The lengths of the bases of a trapezoid are \(x+10\) and \(3 x+2\) and the length of the altitude is 2\(x\) . If the area of the trapezoid is \(40,\) find the lengths of the bases and of the altitude.

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ 2 x^{2}-6 x+3=0 $$

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=2 x^{2}-6 x+7} \\ {y=x+4}\end{array} $$

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ f(x)=2 x^{3}+3 x^{2}-1 \text { and } a=\frac{1}{2} $$

One of the roots is given. Find the other root. \(m^{2}-4 m+n=0 ; 3\)

When \(b^{2}-4 a c=0,\) is \(a x^{2}+b x+c\) a perfect square trinomial or a constant times a perfect square trinomial? Explain why or why not.

In \(18-25,\) write the complex conjugate of each number. $$ \frac{5}{3}-\frac{2}{3} i $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ 2 \sqrt{-5}+\sqrt{-125} $$

The altitude, \(\overline{C D}\) , to the hypotenuse, \(\overline{A B}\) , of right triangle \(A B C\) separates the hypotenuse into two segments, \(\overline{A D}\) and \(\overline{D B}\) . If \(A D=D B+4\) and \(C D=12\) centimeters, find \(D B, A D,\) and \(A B\) . Recall that the length of the altitude to the hypotenuse of a right triangle is the mean proportional between the lengths of the segments into which the hypotenuse is separated, that is, \(\frac{A D}{C D}=\frac{C D}{D B}\)

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ 4 x^{2}-20 x+9=0 $$

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{x^{2}+x-y=7} \\ {\frac{1}{2} x=y+2}\end{array} $$

Find a value of \(c\) such that the roots of \(x^{2}+2 x+c=0\) are: a. equal and rational. b. unequal and rational. c. unequal and irrational. d. not real numbers.

One of the roots is given. Find the other root. \(z^{2}+2 z+k=0 ; \frac{3}{4}\)

In \(18-25,\) write the complex conjugate of each number. $$ \pi+2 i $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{4}+\sqrt{-4}+\sqrt{-36}-\sqrt{36} $$

In \(19-25,\) express each answer in simplest radical form. Check each answer. A parabola is symmetric under a line reflection. Each real root of the quadratic function \(y=a x^{2}+b x+c\) is the image of the other under a reflection in the axis of symmetry of the parabola. a. What are the coordinates of the points at which the parabola whose equation is \(y=a x^{2}+b x+c\) intersects the \(x\) -axis? b. What are the coordinates of the midpoint of the points whose coordinates were foundin part a? c. What is the equation of the axis of symmetry of the parabola \(y=a x^{2}+b x+c ?\) d. The turning point of a parabola is on the axis of symmetry. What is the \(x\) -coordinate of the turning point of the parabola \(y=a x^{2}+b x+c ?\)

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ \frac{1}{2} x^{2}+x-3=0 $$

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=4 x^{2}-6 x-10} \\ {y=25-2 x}\end{array} $$

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ \mathrm{f}(x)=x^{4}-2 x^{2}+x \text { and } a=\sqrt{3} $$

Find a value of \(b\) such that the roots of \(x^{2}+b x+4=0\) are: a. equal and rational. b. unequal and rational. c. unequal and irrational. d. not real numbers.

One of the roots is given. Find the other root. \(x^{2}+b x+3=0 ; 1\)

In \(26-37,\) find each product. $$ (1+5 i)(1+2 i) $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{50}+\sqrt{-2}+\sqrt{200}+\sqrt{-50} $$

Gravity on the moon is about one-sixth of gravity on Earth. An astronaut standing on a tower 20 feet above the moon's surface throws a ball upward with a velocity of 30 feet per second. The height of the ball at any time \(t\) (in seconds) is \(\mathrm{h}(t)=-2.67 t^{2}+30 t+20\) To the nearest tenth of a second, how long will it take for the ball to hit the ground?

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ \frac{1}{4} x^{2}+\frac{3}{4} x-\frac{3}{2}=0 $$

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=x^{2}-2 x+1} \\ {y=-\frac{9 x-35}{2}}\end{array} $$

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ f(x)=x^{3}-2 x+3 \text { and } a=2+i $$

Lauren wants to fence off a rectangular flower bed with a perimeter of 30 yards and a diagonal length of 8 yards. Use the discriminant to determine if her fence can be constructed. If possible, determine the dimensions of the rectangle.