Chapter 5

Without solving each equation, find the sum and product of the roots. \(x^{2}+2 x=0\)

In \(9-14\) : a. For each given value of the discriminant of a quadratic equation with rational coefficients, 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 your answer to part a to determine the number of \(x\) -intercepts of the graph of the corresponding quadratic function. 49

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ (1+0 i)-\left(\frac{1}{5}-\frac{2}{5} i\right) $$

In \(3-18,\) write each number in terms of \(i\) $$ \sqrt{-500} $$

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ 2 x^{2}=x+4 $$

In \(9-14 :\) a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form. $$ f(x)=x^{2}-10 x+18 $$

In \(9-17,\) graph each system and determine the common solution from the graph. $$ \begin{array}{l}{x^{2}-4 x-y+4=0} \\ {y=\frac{4 x+7}{4}}\end{array} $$

In \(3-18,\) find all roots of each given function by factoring or by using the quadratic formula. $$ f(x)=x^{3}-18 x $$

Without solving each equation, find the sum and product of the roots. \(8 x^{2}-9=-6 x\)

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. $$ x^{2}-12 x+36=0 $$

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ \left(-\frac{3}{2}+\frac{5}{3} i\right)-\left(\frac{9}{4}-\frac{1}{5} i\right) $$

In \(3-18,\) write each number in terms of \(i\) $$ 5+\sqrt{-5} $$

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ x^{2}-6 x+3=0 $$

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

In \(3-18,\) find all roots of each given function by factoring or by using the quadratic formula. $$ \mathrm{f}(x)=\left(2 x^{2}+x-1\right)\left(x^{2}-3 x+4\right) $$

Without solving each equation, find the sum and product of the roots. \(x^{2}+2 x-3=0\)

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. $$ 2 x^{2}+7 x=0 $$

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ \left(\frac{1}{4}+12 i\right)+\left(7+\frac{i}{10}\right) $$

In \(3-18,\) write each number in terms of \(i\) $$ 1+\sqrt{-3} $$

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ 4 x^{2}-4 x=11 $$

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ x^{2}+6 x+4=0 $$

In \(9-17,\) graph each system and determine the common solution from the graph. $$ \begin{array}{l}{\frac{y}{x}=\frac{x+7}{5}} \\ {y=2 x}\end{array} $$

In \(3-18,\) find all roots of each given function by factoring or by using the quadratic formula. $$ \mathrm{f}(x)=\left(x^{2}-1\right)\left(3 x^{2}+2 x+1\right) $$

Without solving each equation, find the sum and product of the roots. \(\frac{3 x^{2}+3 x+5}{3}=0\)

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. $$ x^{2}+3 x+1=0 $$

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ \left(\frac{3}{7}+\frac{1}{3} i\right)+\left(\frac{1}{6}-\frac{7}{8} i\right) $$

In \(3-18,\) write each number in terms of \(i\) $$ -4-\sqrt{-24} $$

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ 3 x^{2}=4 x+2 $$

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

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

In \(3-18,\) find all roots of each given function by factoring or by using the quadratic formula. $$ f(x)=x^{3}+2 x^{2}-x-2 $$

One of the roots is given. Find the other root. \(x^{2}+15 x+c=0 ;-5\)

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. $$ 2 x^{2}-8=0 $$

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

In \(3-18,\) write each number in terms of \(i\) $$ -3+2 \sqrt{-9} $$

a. Sketch the graph of \(y=x^{2}+6 x+2\) b. From the graph, estimate the roots of the function to the nearest tenth. c. Use the quadratic formula to find the exact values of the roots of the function. d. Express the roots of the function to the nearest tenth and compare these values to your estimate from the graph.

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ x^{2}+2 x-5=0 $$

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=x^{2}+4 x} \\ {2 x-y=1}\end{array} $$

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

One of the roots is given. Find the other root. \(x^{2}-8 x+c=0 ;-3\)

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. $$ 4 x^{2}-x=1 $$

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

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{-100}+\sqrt{-81} $$

In \(19-25,\) express each answer in simplest radical form. Check each answer. The larger of two numbers is 5 more than twice the smaller. The square of the smaller is equal to the larger. Find the numbers.

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

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

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

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. $$ 3 x^{2}=5 x-3 $$

In \(18-25,\) write the complex conjugate of each number. $$ -8+i $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{-25}-\sqrt{-4} $$