Chapter 11

An equation in the form \(a x^{2}+b x+c=0,\) where \(a, b,\) and \(c\) are real numbers and \(a \neq 0,\) is \(\mathrm{a}(\mathrm{n})\) _________ equation, also called a(n) ________ degree equation. The greatest degree of the variable is _____.

Which step is an appropriate way to begin solving the quadratic equation \(2 x^{2}-4 x=9\) by completing the square? A. Add 4 to each side. B. Factor the left side as \(2 x(x-2)\) C. Factor the left side as \(x(2 x-4)\). D. Divide each side by 2 .

Which of the following are quadratic equations? A. \(x+2 y=0\) B. \(x^{2}-8 x+16=0\) C. \(2 x^{2}-5 x=3\) D. \(x^{3}+x^{2}+4=0\)

What is the first step in solving a formula like \(g w^{2}=2 r\) for \(w ?\)

Which step is an appropriate way to begin solving the quadratic equation \(x^{2}+12 x=13\) by completing the square? A. Add 36 to each side. B. Subtract 13 from each side. C. Divide each side by \(x\). D. Add 6 to each side.

A student solved \(5 x^{2}-5 x+1=0\) incorrectly as follows. $$ \begin{array}{l} x=\frac{-(-5) \pm \sqrt{(-5)^{2}-4(5)(1)}}{2(5)} \\ x=\frac{5 \pm \sqrt{5}}{10} \\ x=\frac{1}{2} \pm \sqrt{5} \end{array} $$

How can we determine the number of \(x\) -intercepts of the graph of a quadratic function without graphing the function?

Match each quadratic function in Column I with the description of the parabola that is its graph in Column II. I (a) \(f(x)=(x-4)^{2}-2\) (b) \(f(x)=(x-2)^{2}-4\) (c) \(f(x)=(x+4)^{2}+2\) (d) \(f(x)=-(x-4)^{2}-2\) (e) \(f(x)=-(x-2)^{2}-4\) (f) \(f(x)=-(x-4)^{2}+2\) II A. Vertex \((2,-4),\) opens down B. Vertex \((2,-4),\) opens up C. Vertex \((4,-2),\) opens down D. Vertex \((4,-2),\) opens up E. Vertex \((4,2),\) opens down F. Vertex \((-4,2),\) opens up

Which quadratic equation is in standard form? A. \(x^{2}=25\) B. \(3 x^{2}-x=4\) C. \((x-5)^{2}=16\) D. \(x^{2}-x-2=0\)

Based on the discussion and examples of this section, give the first step to solve each equation. Do not actually solve. \(\left(x^{2}+x\right)^{2}-8\left(x^{2}+x\right)+12=0\)

Give the correct solution set. A student incorrectly claimed that the equation \(2 x^{2}-5=0\) cannot be solved using the quadratic formula because there is no first-degree x-term.

For the quadratic function \(f(x)=a(x-h)^{2}+k,\) in what quadrant is the vertex if the values of \(h\) and \(k\) are as follows? (a) \(h>0, k>0\) (b) \(h>0, k<0\) (c) \(h<0, k>0\) (d) \(h<0, k<0\) Consider the value of \(a\), and make the correct choice. (e) If \(a>0,\) then the graph opens (up/down). (f) If \(a<0,\) then the graph opens (up/down). (g) If \(|a|>1,\) then the graph is (narrower / wider) than the graph of \(f(x)=x^{2}\). (h) If \(0<|a|<1,\) then the graph is (narrower / wider) than the graph of \(f(x)=x^{2}\).

The equation \(x^{2}=-9\) has solutions that (are/are not) real numbers. These solutions involve the imaginary unit \(i,\) which is defined as \(i=\) ________ Thus, \(i^{2}=\) ____________ For any positive real number \(a\), we have \(\sqrt{-a}=\) __________.

Why is it particularly important to check all proposed solutions to an applied problem against the information in the original problem?

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ x^{2}-8 x+15=0 $$

Identify the vertex of each parabola. $$ f(x)=-3 x^{2} $$

Match each equation in Column I with the correct description of its solution in Column II. I (a) \(x^{2}=12\) (b) \(x^{2}=-16\) (c) \(x^{2}=\frac{25}{36}\) (d) \(x^{2}=100\) II A. Two nonreal complex B. Two integer solutions solutions C. Two irrational solutions D. Two rational solutions that are not integers

Which equations have a graph that is a vertical parabola? A horizontal parabola? A. \(y=-x^{2}+20 x+80\) B. \(x=2 y^{2}+6 y+5\) C. \(x+1=(y+2)^{2}\) D. \(f(x)=(x-4)^{2}\)

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ x^{2}+3 x-28=0 $$

A student incorrectly solved the following equation as shown. $$ \begin{array}{r} x^{2}-x-2=4 \\ (x-2)(x+1)=4 \\ x-2=4 \text { or } x+1=4 \\ x=6 \quad \text { or } \quad x=3 \end{array} $$ Solve correctly and give the solution set.

Identify the vertex of each parabola. $$ f(x)=-4 x^{2} $$

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ 6 x^{2}+11 x-10=0 $$

A student solving \(x^{2}=81\) wrote the solution set incorrectly as \(\\{9\\} .\) Her teacher did not give her full credit. The student argued that because \(9^{2}=81,\) her answer had to be correct. Give the correct solution set.

Identify the vertex of each parabola. $$ f(x)=\frac{1}{3} x^{2} $$

Solve each equation. Check the solutions. \(\frac{14}{x}=x-5\)

Find the vertex of each parabola. $$ f(x)=x^{2}+8 x+10 $$

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ 8 x^{2}+10 x-3=0 $$

When solving a quadratic equation, a student obtained the solutions \(x=\frac{3+2 \sqrt{5}}{2}\) or \(x=\frac{3-2 \sqrt{5}}{2}\) and he wrote the solution set incorrectly as \(\\{3 \pm \sqrt{5}\\}\). Give the correct solution set.

Identify the vertex of each parabola. $$ f(x)=\frac{1}{2} x^{2} $$

Solve each equation. Check the solutions. \(\frac{-12}{x}=x+8\)

Find the vertex of each parabola. $$ f(x)=x^{2}+10 x+23 $$

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ 4 x^{2}+12 x+9=0 $$

Solve using the zero-factor property. $$ x^{2}-x-56=0 $$

Identify the vertex of each parabola. $$ f(x)=x^{2}+4 $$

Solve each equation. Check the solutions. \(-\frac{3}{x}-\frac{28}{x^{2}}=0\)

Find the vertex of each parabola. $$ f(x)=-2 x^{2}+4 x-5 $$

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ 16 x^{2}+40 x+25=0 $$

Solve using the zero-factor property. $$ x^{2}-2 x-99=0 $$

Identify the vertex of each parabola. $$ f(x)=x^{2}-4 $$

Solve each equation. Check the solutions. \(4-\frac{7}{r}-\frac{2}{r^{2}}=0\)

Find the vertex of each parabola. $$ f(x)=-3 x^{2}+12 x-8 $$

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ 36 x^{2}-12 x+1=0 $$

Solve using the zero-factor property. $$ x^{2}-8 x+15=0 $$

Identify the vertex of each parabola. $$ f(x)=(x-1)^{2} $$

Solve each equation. Check the solutions. \(3-\frac{1}{t}=\frac{2}{t^{2}}\)

Find the vertex of each parabola. $$ f(x)=x^{2}+x-7 $$

Solve each formula for the specified variable. (Leave \(\pm\) in the answers as needed.) See Examples I and 2. \(I=\frac{k s}{d^{2}}\) for \(d\)

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ 9 x^{2}-6 x+1=0 $$

Solve using the zero-factor property. $$ x^{2}-6 x+5=0 $$

Identify the vertex of each parabola. $$ f(x)=(x+3)^{2} $$