Chapter 5

Explain the relationship between the solutions of \(y>a x^{2}+b x+c\) and the solutions of \(0>a x^{2}+b x+c\)

Sharon said that if \(\mathrm{f}(x)\) is a polynomial function and \(\mathrm{f}(a)=0,\) then \(a\) is a root of the function. Do you agree with Sharon? Explain why or why not.

Emily said that when \(a\) and \(c\) are real numbers with the same sign and \(b=0\) , the roots of the equation \(a x^{2}+b x+c=0\) are pure imaginary. Do you agree with Emily? Justify your answer.

a. What is the discriminant of the equation \(x^{2}+(\sqrt{5}) x-1=0 ?\) b. Find the roots of the equation \(x^{2}+(\sqrt{5}) x-1=0\) c. Do the rules for the relationship between the discriminant and the roots of the equation given in this chapter apply to this equation? Explain why or why not.

The roots of a quadratic equation with rational coefficients are \(p \pm \sqrt{q} .\) Write the equation in standard form in terms of \(p\) and \(q .\)

Tim said that the binomial \(x^{2}+16\) can be written as \(x^{2}-16 i^{2}\) and factored over the set of complex numbers. Do you agree with Tim? Explain why or why not.

Pete said that \(\sqrt{-2} \times \sqrt{-8}=\sqrt{16}=4 .\) Do you agree with Pete? Explain why or why not.

Noah said that the solutions of Example \(1, \frac{2+\sqrt{6}}{2}\) and \(\frac{2-\sqrt{6}}{2},\) could have been written as \(\frac{2+\sqrt{6}}{2}=1+\sqrt{6}\) and \(\frac{-}{2}-\sqrt{6}=1-\sqrt{6} .\) Do you agree with Noah? Explain why or why not.

Explain why the equations \(y=x^{2}+2\) and \(y=-2\) have no common solution in the set of real numbers.

Noah said that if \(a, b,\) and \(c\) are rational numbers and \(b^{2}-4 a c<0\) , then the roots of the equation \(a x^{2}+b x+c=0\) are complex conjugates. Do you agree with Noah? Justify your answer.

Jordan said that if the roots of a polynomial function \(\mathrm{f}(x)\) are \(r_{1}, r_{2},\) and \(r_{3},\) then the roots of \(\mathrm{g}(x)=\mathrm{f}(x-a)\) are \(r_{1}+a, r_{2}+a,\) and \(r_{3}+a .\) Do you agree with Jordan? Explain why or why not.

Christina said that when \(a, b,\) and \(c\) are rational numbers and \(a\) and \(c\) have opposite signs, the quadratic equation \(a x^{2}+b x+c=0\) must have real roots. Do you agree with Christina? Explain why or why not.

Adrien said that if the roots of a quadratic equation are \(\frac{1}{2}\) and \(\frac{3}{4}\) , the equation is \(4 x^{2}-5 x+\frac{3}{2}=0 .\) Olivia said that the equation is \(8 x^{2}-10 x+3=0 .\) Who is correct? Justify your answer.

Joshua said that the product of a complex number and its conjugate is always a real number. Do you agree with Joshua? Explain why or why not.

Ethan said that the square of any pure imaginary number is a negative real number. Do you agree with Ethan? Justify your answer.

Rita said that when \(a, b,\) and \(c\) are real numbers, the roots of \(a x^{2}+b x+c=0\) are real numbers only when \(b^{2} \geq 4 a c .\) Do you agree with Rita? Explain why or why not.

Phillip said that the equation \(0=x^{2}-6 x+1\) can be solved by adding 8 to both sides of the equation. Do you agree with Phillip? Explain why or why not.

In \(3-14,\) use the quadratic formula to find the imaginary roots of each equation. $$ x^{2}-4 x+8=0 $$

In \(3-18,\) find all roots of each given function by factoring or by using the quadratic formula. $$ \mathrm{f}(x)=x^{3}+7 x^{2}+10 x $$

Without solving each equation, find the sum and product of the roots. \(x^{2}+x+1=0\)

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ (6+7 i)+(1+2 i) $$

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

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ x^{2}+5 x+4=0 $$

In \(3-8,\) complete the square of the quadratic expression. $$ x^{2}+6 x $$

In \(3-14,\) use the quadratic formula to find the imaginary roots of each equation. $$ x^{2}+6 x+10=0 $$

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

Without solving each equation, find the sum and product of the roots. \(x^{2}+4 x+5=0\)

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ (3-5 i)+(2+i) $$

In \(3-18,\) write each number in terms of \(i\) $$ \sqrt{-81} $$

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ x^{2}+6 x-7=0 $$

In \(3-8,\) complete the square of the quadratic expression. $$ x^{2}-8 x $$

In \(3-14,\) use the quadratic formula to find the imaginary roots of each equation. $$ x^{2}-4 x+13=0 $$

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

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ (5-6 i)+(4+2 i) $$

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

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ x^{2}-3 x+1=0 $$

In \(3-8,\) complete the square of the quadratic expression. $$ x^{2}-2 x $$

In \(3-14,\) use the quadratic formula to find the imaginary roots of each equation. $$ 2 x^{2}+2 x+1=0 $$

Without solving each equation, find the sum and product of the roots. \(5 x^{2}+2 x-10=0\)

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ (-3+3 i)-(1+5 i) $$

In \(3-18,\) write each number in terms of \(i\) $$ -\sqrt{-36} $$

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ x^{2}-x-4=0 $$

In \(3-8,\) complete the square of the quadratic expression. $$ x^{2}-12 x $$

In \(3-14,\) use the quadratic formula to find the imaginary roots of each equation. $$ x^{2}+10 x+29=0 $$

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

Without solving each equation, find the sum and product of the roots. \(3 x^{2}-6 x+4=0\)

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ (2-8 i)-(2+8 i) $$

In \(3-18,\) write each number in terms of \(i\) $$ -\sqrt{-121} $$

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ x^{2}+5 x-2=0 $$

In \(3-8,\) complete the square of the quadratic expression. $$ 2 x^{2}-4 x $$