Chapter 9 Review and Practice Test

Learning objectives

Learning objectives

• Review the key ideas of this chapter.

• Apply the chapter's methods to review exercises.

• Check your understanding with a practice test.

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Worked example

discriminant

b

2

ac

quadratic function

a

b

c

quadratic inequality

A quadratic inequality is an inequality that contains a quadratic expression.

##

Key Concepts

• Square Root Property If ${x}^{2}=k$, then $x=\sqrt{k}\,\mathrm{or}\,x=-\sqrt{k}$ or $x=\pm\sqrt{k}$ How to solve a quadratic equation using the square root property. Step 1. Isolate the quadratic term and make its coefficient one. Step 2. Use Square Root Property. Step 3. Simplify the radical. Step 4. Check the solutions.

• Binomial Squares Pattern If a and b are real numbers,

• How to Complete a Square Step 1. Identify b, the coefficient of x. Step 2. Find ${(\frac{1}{2}b)}^{2},$ the number to complete the square. Step 3. Add the ${(\frac{1}{2}b)}^{2}$ to x${}^{2}$ + bx Step 4. Rewrite the trinomial as a binomial square

• How to solve a quadratic equation of the form ax${}^{2}$ + bx + c = 0 by completing the square. Step 1. Divide by a to make the coefficient of x${}^{2}$ term 1. Step 2. Isolate the variable terms on one side and the constant terms on the other. Step 3. Find ${(\frac{1}{2}\,\cdot\,b)}^{2},$ the number needed to complete the square. Add it to both sides of the equation. Step 4. Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right. Step 5. Use the Square Root Property. Step 6. Simplify the radical and then solve the two resulting equations. Step 7. Check the solutions.

• Quadratic Formula The solutions to a quadratic equation of the form ax${}^{2}$ + bx + c = 0, $a\ne{}0$ are given by the formula:

• How to solve a quadratic equation using the Quadratic Formula. Step 1. Write the quadratic equation in standard form, ax${}^{2}$ + bx + c = 0. Identify the values of a, b, c. Step 2. Write the Quadratic Formula. Then substitute in the values of a, b, c. Step 3. Simplify. Step 4. Check the solutions.

• Using the Discriminant, b${}^{2}$ − 4*ac*, to Determine the Number and Type of Solutions of a Quadratic Equation For a quadratic equation of the form ax${}^{2}$ + bx + c = 0, $a\ne{}0,$ If b${}^{2}$ − 4*ac* > 0, the equation has 2 real solutions. if b${}^{2}$ − 4*ac* = 0, the equation has 1 real solution. if b${}^{2}$ − 4*ac* < 0, the equation has 2 complex solutions.

• Methods to Solve Quadratic Equations: Factoring Square Root Property Completing the Square Quadratic Formula

• How to identify the most appropriate method to solve a quadratic equation. Step 1. Try Factoring first. If the quadratic factors easily, this method is very quick. Step 2. Try the Square Root Property next. If the equation fits the form ax${}^{2}$ = k or a(x − h)${}^{2}$ = k, it can easily be solved by using the Square Root Property. Step 3. Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

• How to solve equations in quadratic form. Step 1. Identify a substitution that will put the equation in quadratic form. Step 2. Rewrite the equation with the substitution to put it in quadratic form. Step 3. Solve the quadratic equation for u. Step 4. Substitute the original variable back into the results, using the substitution. Step 5. Solve for the original variable. Step 6. Check the solutions.

• Methods to Solve Quadratic Equations Factoring Square Root Property Completing the Square Quadratic Formula

• How to use a Problem-Solving Strategy. Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Area of a Triangle For a triangle with base, b, and height, h, the area, A, is given by the formula $A=\frac{1}{2}bh.$

• Area of a Rectangle For a rectangle with length, L, and width, W, the area, A, is given by the formula A = LW.

• Pythagorean Theorem In any right triangle, where a and b are the lengths of the legs, and c is the length of the hypotenuse, a${}^{2}$ + b${}^{2}$ = c${}^{2}$.

• Projectile motion The height in feet, h, of an object shot upwards into the air with initial velocity, v${}_{0}$, after t seconds is given by the formula h = −16*t*${}^{2}$ + v${}_{0}$t.

• Parabola Orientation For the graph of the quadratic function $f(x)=a{x}^{2}+bx+c,$ if a > 0, the parabola opens upward. a < 0, the parabola opens downward.

• Axis of Symmetry and Vertex of a Parabola The graph of the function $f(x)=a{x}^{2}+bx+c$ is a parabola where: the axis of symmetry is the vertical line $x=-\frac{b}{2a}.$ the vertex is a point on the axis of symmetry, so its x-coordinate is $-\frac{b}{2a}.$ the y-coordinate of the vertex is found by substituting $x=-\frac{b}{2a}$ into the quadratic equation.

• Find the Intercepts of a Parabola To find the intercepts of a parabola whose function is $f(x)=a{x}^{2}+bx+c:$

• How to graph a quadratic function using properties. Step 1. Determine whether the parabola opens upward or downward. Step 2. Find the equation of the axis of symmetry. Step 3. Find the vertex. Step 4. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry. Step 5. Find the x-intercepts. Find additional points if needed. Step 6. Graph the parabola.

• Minimum or Maximum Values of a Quadratic Equation The y-coordinate of the vertex of the graph of a quadratic equation is the minimum value of the quadratic equation if the parabola opens upward. maximum value of the quadratic equation if the parabola opens downward.

• Graph a Quadratic Function of the form $f(x)={x}^{2}+k$ Using a Vertical Shift The graph of $f(x)={x}^{2}+k$ shifts the graph of $f(x)={x}^{2}$ vertically k units. If k > 0, shift the parabola vertically up k units. If k < 0, shift the parabola vertically down $|k|$ units.

• Graph a Quadratic Function of the form $f(x)={(x-h)}^{2}$ Using a Horizontal Shift The graph of $f(x)={(x-h)}^{2}$ shifts the graph of $f(x)={x}^{2}$ horizontally h units. If h > 0, shift the parabola horizontally left h units. If h < 0, shift the parabola horizontally right $|h|$ units.

• Graph of a Quadratic Function of the form $f(x)=a{x}^{2}$ The coefficient a in the function $f(x)=a{x}^{2}$ affects the graph of $f(x)={x}^{2}$ by stretching or compressing it. If $0<|a|<1,$ then the graph of $f(x)=a{x}^{2}$ will be “wider” than the graph of $f(x)={x}^{2}.$ If $|a|>1,$ then the graph of $f(x)=a{x}^{2}$ will be “skinnier” than the graph of $f(x)={x}^{2}.$

• How to graph a quadratic function using transformations Step 1. Rewrite the function in $f(x)=a{(x-h)}^{2}+k$ form by completing the square. Step 2. Graph the function using transformations.

• Graph a quadratic function in the vertex form $f(x)=a{(x-h)}^{2}+k$ using properties Step 1. Rewrite the function in $f(x)=a{(x-h)}^{2}+k$ form. Step 2. Determine whether the parabola opens upward, a > 0, or downward, a < 0. Step 3. Find the axis of symmetry, x = h. Step 4. Find the vertex, (h, k). Step 5. Find the*y*-intercept. Find the point symmetric to the y-intercept across the axis of symmetry. Step 6. Find the x-intercepts, if possible. Step 7. Graph the parabola.

• Solve a Quadratic Inequality Graphically Step 1. Write the quadratic inequality in standard form. Step 2. Graph the function $f(x)=a{x}^{2}+bx+c$ using properties or transformations. Step 3. Determine the solution from the graph.

• How to Solve a Quadratic Inequality Algebraically Step 1. Write the quadratic inequality in standard form. Step 2. Determine the zero partition numbers -- the solutions to the related quadratic equation. Step 3. Use the zero partition numbers to divide the number line into intervals. Step 4. Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality. Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.

##

Review Exercises

Solve Quadratic Equations of the form ax${}^{2}$ = k Using the Square Root Property

In the following exercises, solve using the Square Root Property.

Solve Quadratic Equations of the Form $a{(x-h)}^{2}=k$ Using the Square Root Property

In the following exercises, solve using the Square Root Property.

Solve Quadratic Equations Using Completing the Square

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

In the following exercises, solve by completing the square.

Solve Quadratic Equations of the form ax${}^{2}$ + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

In the following exercises, solve by using the Quadratic Formula.

Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

In the following exercises, determine the number of solutions for each quadratic equation.

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

Solve Equations in Quadratic Form

In the following exercises, solve.

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

Recognize the Graph of a Quadratic Function

In the following exercises, graph by plotting point.

In the following exercises, determine if the following parabolas open up or down.

Find the Axis of Symmetry and Vertex of a Parabola

In the following exercises, find ⓐ the equation of the axis of symmetry and ⓑ the vertex.

Find the Intercepts of a Parabola

In the following exercises, find the x- and y-intercepts.

Graph Quadratic Functions Using Properties

In the following exercises, graph by using its properties.

Solve Maximum and Minimum Applications

In the following exercises, find the minimum or maximum value.

In the following exercises, solve. Rounding answers to the nearest tenth.

Graph Quadratic Functions of the form $f(x)={x}^{2}+k$

In the following exercises, graph each function using a vertical shift.

In the following exercises, graph each function using a horizontal shift.

In the following exercises, graph each function using transformations.

Graph Quadratic Functions of the form $f(x)=a{x}^{2}$

In the following exercises, graph each function.

Graph Quadratic Functions Using Transformations

In the following exercises, rewrite each function in the $f(x)=a{(x-h)}^{2}+k$ form by completing the square.

In the following exercises, ⓐ rewrite each function in $f(x)=a{(x-h)}^{2}+k$ form and ⓑ graph it by using transformations.

In the following exercises, ⓐ rewrite each function in $f(x)=a{(x-h)}^{2}+k$ form and ⓑ graph it using properties.

Find a Quadratic Function from its Graph

In the following exercises, write the quadratic function in $f(x)=a{(x-h)}^{2}+k$ form.

Solve Quadratic Inequalities Graphically

In the following exercises, solve graphically and write the solution in interval notation.

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

##

Solve the following quadratic equations. Use any method.

Use the discriminant to determine the number and type of solutions of each quadratic equation.

Solve each equation.

For each parabola, find ⓐ which direction it opens, ⓑ the equation of the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and e) the maximum or minimum value.

Graph each quadratic function using intercepts, the vertex, and the equation of the axis of symmetry.

In the following exercises, graph each function using transformations.

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

Model the situation with a quadratic equation and solve by any method.

Key Terms

discriminant

b

2

ac

quadratic function

a

b

c

quadratic inequality

A quadratic inequality is an inequality that contains a quadratic expression.

##

Key Concepts

• Square Root Property If ${x}^{2}=k$, then $x=\sqrt{k}\,\mathrm{or}\,x=-\sqrt{k}$ or $x=\pm\sqrt{k}$ How to solve a quadratic equation using the square root property. Step 1. Isolate the quadratic term and make its coefficient one. Step 2. Use Square Root Property. Step 3. Simplify the radical. Step 4. Check the solutions.

• Binomial Squares Pattern If a and b are real numbers,

• How to Complete a Square Step 1. Identify b, the coefficient of x. Step 2. Find ${(\frac{1}{2}b)}^{2},$ the number to complete the square. Step 3. Add the ${(\frac{1}{2}b)}^{2}$ to x${}^{2}$ + bx Step 4. Rewrite the trinomial as a binomial square

• How to solve a quadratic equation of the form ax${}^{2}$ + bx + c = 0 by completing the square. Step 1. Divide by a to make the coefficient of x${}^{2}$ term 1. Step 2. Isolate the variable terms on one side and the constant terms on the other. Step 3. Find ${(\frac{1}{2}\,\cdot\,b)}^{2},$ the number needed to complete the square. Add it to both sides of the equation. Step 4. Factor the perfect square trinomial, writing it as a binomial squared on the left and simplify by adding the terms on the right. Step 5. Use the Square Root Property. Step 6. Simplify the radical and then solve the two resulting equations. Step 7. Check the solutions.

• Quadratic Formula The solutions to a quadratic equation of the form ax${}^{2}$ + bx + c = 0, $a\ne{}0$ are given by the formula:

• How to solve a quadratic equation using the Quadratic Formula. Step 1. Write the quadratic equation in standard form, ax${}^{2}$ + bx + c = 0. Identify the values of a, b, c. Step 2. Write the Quadratic Formula. Then substitute in the values of a, b, c. Step 3. Simplify. Step 4. Check the solutions.

• Using the Discriminant, b${}^{2}$ − 4*ac*, to Determine the Number and Type of Solutions of a Quadratic Equation For a quadratic equation of the form ax${}^{2}$ + bx + c = 0, $a\ne{}0,$ If b${}^{2}$ − 4*ac* > 0, the equation has 2 real solutions. if b${}^{2}$ − 4*ac* = 0, the equation has 1 real solution. if b${}^{2}$ − 4*ac* < 0, the equation has 2 complex solutions.

• Methods to Solve Quadratic Equations: Factoring Square Root Property Completing the Square Quadratic Formula

• How to identify the most appropriate method to solve a quadratic equation. Step 1. Try Factoring first. If the quadratic factors easily, this method is very quick. Step 2. Try the Square Root Property next. If the equation fits the form ax${}^{2}$ = k or a(x − h)${}^{2}$ = k, it can easily be solved by using the Square Root Property. Step 3. Use the Quadratic Formula. Any other quadratic equation is best solved by using the Quadratic Formula.

• How to solve equations in quadratic form. Step 1. Identify a substitution that will put the equation in quadratic form. Step 2. Rewrite the equation with the substitution to put it in quadratic form. Step 3. Solve the quadratic equation for u. Step 4. Substitute the original variable back into the results, using the substitution. Step 5. Solve for the original variable. Step 6. Check the solutions.

• Methods to Solve Quadratic Equations Factoring Square Root Property Completing the Square Quadratic Formula

• How to use a Problem-Solving Strategy. Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

• Area of a Triangle For a triangle with base, b, and height, h, the area, A, is given by the formula $A=\frac{1}{2}bh.$

• Area of a Rectangle For a rectangle with length, L, and width, W, the area, A, is given by the formula A = LW.

• Pythagorean Theorem In any right triangle, where a and b are the lengths of the legs, and c is the length of the hypotenuse, a${}^{2}$ + b${}^{2}$ = c${}^{2}$.

• Projectile motion The height in feet, h, of an object shot upwards into the air with initial velocity, v${}_{0}$, after t seconds is given by the formula h = −16*t*${}^{2}$ + v${}_{0}$t.

• Parabola Orientation For the graph of the quadratic function $f(x)=a{x}^{2}+bx+c,$ if a > 0, the parabola opens upward. a < 0, the parabola opens downward.

• Axis of Symmetry and Vertex of a Parabola The graph of the function $f(x)=a{x}^{2}+bx+c$ is a parabola where: the axis of symmetry is the vertical line $x=-\frac{b}{2a}.$ the vertex is a point on the axis of symmetry, so its x-coordinate is $-\frac{b}{2a}.$ the y-coordinate of the vertex is found by substituting $x=-\frac{b}{2a}$ into the quadratic equation.

• Find the Intercepts of a Parabola To find the intercepts of a parabola whose function is $f(x)=a{x}^{2}+bx+c:$

• How to graph a quadratic function using properties. Step 1. Determine whether the parabola opens upward or downward. Step 2. Find the equation of the axis of symmetry. Step 3. Find the vertex. Step 4. Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry. Step 5. Find the x-intercepts. Find additional points if needed. Step 6. Graph the parabola.

• Minimum or Maximum Values of a Quadratic Equation The y-coordinate of the vertex of the graph of a quadratic equation is the minimum value of the quadratic equation if the parabola opens upward. maximum value of the quadratic equation if the parabola opens downward.

• Graph a Quadratic Function of the form $f(x)={x}^{2}+k$ Using a Vertical Shift The graph of $f(x)={x}^{2}+k$ shifts the graph of $f(x)={x}^{2}$ vertically k units. If k > 0, shift the parabola vertically up k units. If k < 0, shift the parabola vertically down $|k|$ units.

• Graph a Quadratic Function of the form $f(x)={(x-h)}^{2}$ Using a Horizontal Shift The graph of $f(x)={(x-h)}^{2}$ shifts the graph of $f(x)={x}^{2}$ horizontally h units. If h > 0, shift the parabola horizontally left h units. If h < 0, shift the parabola horizontally right $|h|$ units.

• Graph of a Quadratic Function of the form $f(x)=a{x}^{2}$ The coefficient a in the function $f(x)=a{x}^{2}$ affects the graph of $f(x)={x}^{2}$ by stretching or compressing it. If $0<|a|<1,$ then the graph of $f(x)=a{x}^{2}$ will be “wider” than the graph of $f(x)={x}^{2}.$ If $|a|>1,$ then the graph of $f(x)=a{x}^{2}$ will be “skinnier” than the graph of $f(x)={x}^{2}.$

• How to graph a quadratic function using transformations Step 1. Rewrite the function in $f(x)=a{(x-h)}^{2}+k$ form by completing the square. Step 2. Graph the function using transformations.

• Graph a quadratic function in the vertex form $f(x)=a{(x-h)}^{2}+k$ using properties Step 1. Rewrite the function in $f(x)=a{(x-h)}^{2}+k$ form. Step 2. Determine whether the parabola opens upward, a > 0, or downward, a < 0. Step 3. Find the axis of symmetry, x = h. Step 4. Find the vertex, (h, k). Step 5. Find the*y*-intercept. Find the point symmetric to the y-intercept across the axis of symmetry. Step 6. Find the x-intercepts, if possible. Step 7. Graph the parabola.

• Solve a Quadratic Inequality Graphically Step 1. Write the quadratic inequality in standard form. Step 2. Graph the function $f(x)=a{x}^{2}+bx+c$ using properties or transformations. Step 3. Determine the solution from the graph.

• How to Solve a Quadratic Inequality Algebraically Step 1. Write the quadratic inequality in standard form. Step 2. Determine the zero partition numbers -- the solutions to the related quadratic equation. Step 3. Use the zero partition numbers to divide the number line into intervals. Step 4. Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality. Step 5. Determine the intervals where the inequality is correct. Write the solution in interval notation.

##

Review Exercises

Solve Quadratic Equations of the form ax${}^{2}$ = k Using the Square Root Property

In the following exercises, solve using the Square Root Property.

Solve Quadratic Equations of the Form $a{(x-h)}^{2}=k$ Using the Square Root Property

In the following exercises, solve using the Square Root Property.

Solve Quadratic Equations Using Completing the Square

In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.

In the following exercises, solve by completing the square.

Solve Quadratic Equations of the form ax${}^{2}$ + bx + c = 0 by Completing the Square

In the following exercises, solve by completing the square.

In the following exercises, solve by using the Quadratic Formula.

Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

In the following exercises, determine the number of solutions for each quadratic equation.

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.

Solve Equations in Quadratic Form

In the following exercises, solve.

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.

Recognize the Graph of a Quadratic Function

In the following exercises, graph by plotting point.

In the following exercises, determine if the following parabolas open up or down.

Find the Axis of Symmetry and Vertex of a Parabola

In the following exercises, find ⓐ the equation of the axis of symmetry and ⓑ the vertex.

Find the Intercepts of a Parabola

In the following exercises, find the x- and y-intercepts.

Graph Quadratic Functions Using Properties

In the following exercises, graph by using its properties.

Solve Maximum and Minimum Applications

In the following exercises, find the minimum or maximum value.

In the following exercises, solve. Rounding answers to the nearest tenth.

Graph Quadratic Functions of the form $f(x)={x}^{2}+k$

In the following exercises, graph each function using a vertical shift.

In the following exercises, graph each function using a horizontal shift.

In the following exercises, graph each function using transformations.

Graph Quadratic Functions of the form $f(x)=a{x}^{2}$

In the following exercises, graph each function.

Graph Quadratic Functions Using Transformations

In the following exercises, rewrite each function in the $f(x)=a{(x-h)}^{2}+k$ form by completing the square.

In the following exercises, ⓐ rewrite each function in $f(x)=a{(x-h)}^{2}+k$ form and ⓑ graph it by using transformations.

In the following exercises, ⓐ rewrite each function in $f(x)=a{(x-h)}^{2}+k$ form and ⓑ graph it using properties.

Find a Quadratic Function from its Graph

In the following exercises, write the quadratic function in $f(x)=a{(x-h)}^{2}+k$ form.

Solve Quadratic Inequalities Graphically

In the following exercises, solve graphically and write the solution in interval notation.

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

##

Solve the following quadratic equations. Use any method.

Use the discriminant to determine the number and type of solutions of each quadratic equation.

Solve each equation.

For each parabola, find ⓐ which direction it opens, ⓑ the equation of the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and e) the maximum or minimum value.

Graph each quadratic function using intercepts, the vertex, and the equation of the axis of symmetry.

In the following exercises, graph each function using transformations.

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

Model the situation with a quadratic equation and solve by any method.

1. ${y}^{2}=144$

2. $4{a}^{2}=100$

3. ${r}^{2}+32=0$

4. $\frac{2}{3}{w}^{2}-20=30$

5. ${(p-5)}^{2}+3=19$

1. ${(x-\frac{1}{4})}^{2}=\frac{3}{16}$

2. ${(n-4)}^{2}-50=150$

3. ${n}^{2}+10n+25=12$

4. ${x}^{2}+22x$

5. ${a}^{2}-3a$

1. ${d}^{2}+14d=-13$

2. ${m}^{2}+6m=-109$

3. ${v}^{2}-14v=-31$

4. ${m}^{2}+10m-4=-13$

5. ${a}^{2}=3a+8$

1. $(u+8)(u+4)=14$

2. $3{p}^{2}-18p+15=15$

3. $4{y}^{2}-6y=4$

4. $3{c}^{2}+2c=9$

5. $2{x}^{2}+6x=-5$

1. $4{x}^{2}-5x+1=0$

2. ${r}^{2}-r-42=0$

3. $4{v}^{2}+v-5=0$

4. $3{m}^{2}+8m+2=0$

5. $6{a}^{2}-5a+2=0$

1. $u(u-10)+3=0$

2. $\frac{1}{8}{p}^{2}-\frac{1}{5}p=-\frac{1}{20}$

3. $4{c}^{2}+4c+1=0$

4. ⓐ $9{x}^{2}-6x+1=0$ⓑ $3{y}^{2}-8y+1=0$ⓒ $7{m}^{2}+12m+4=0$ⓓ $5{n}^{2}-n+1=0$

5. ⓐ $16{r}^{2}-8r+1=0$ⓑ $5{t}^{2}-8t+3=9$ⓒ $3{(c+2)}^{2}=15$

1. ${x}^{4}-14{x}^{2}+24=0$

2. $4{x}^{4}-5{x}^{2}+1=0$

3. $x+3\sqrt{x}-28=0$

4. ${x}^{\frac{2}{3}}-10{x}^{\frac{1}{3}}+24=0$

5. $8{x}^{-2}-2{x}^{-1}-3=0$

1. Two handymen can do a home repair in 2 hours if they work together. One of the men takes 3 hours more than the other man to finish the job by himself. How long does it take for each handyman to do the home repair individually?

2. ⓐ $y={x}^{2}+8x-1$ⓑ $y=-4{x}^{2}-7x+1$

3. $y=2{x}^{2}-8x+1$

4. $y={x}^{2}-8x+15$

5. $y=-5{x}^{2}-30x-46$

1. $f(x)=2{x}^{2}+4x+6$

2. $f(x)=-3{x}^{2}-12x-5$

3. ${x}^{2}-x-6>0$

4. $-{x}^{2}-x+2\ge{}0$

5. ${x}^{2}-6x+8<0$

1. ${x}^{2}-6x+4\le{}0$

2. $-{x}^{2}+x-6>0$

3. Use the Square Root Property to solve the quadratic equation $3{(w+5)}^{2}=27.$

4. Use the Quadratic Formula to solve the quadratic equation $2{m}^{2}-5m+3=0.$

5. $\frac{9}{4}{y}^{2}-3y+1=0$

1. $3{q}^{2}-10q+12=0$

2. ${y}^{\frac{2}{3}}+2{y}^{\frac{1}{3}}-3=0$

3. $y=-{x}^{2}-8x-16$

4. $f(x)=-2{x}^{2}+8x+4$

5. $f(x)={x}^{2}-4x-1$

1. Find two consecutive even numbers whose product is 624.

2. Julius built a triangular display case for his coin collection. The height of the display case is six inches less than twice the width of the base. The area of the of the back of the case is 70 square inches. Find the height and width of the case.

3. A rectangular piece of plywood has a diagonal which measures two feet more than the width. The length of the plywood is twice the width. What is the length of the plywood’s diagonal? Round to the nearest tenth.

4. For Sophia’s graduation party, several tables of the same width will be arranged end to end to give serving table with a total area of 75 square feet. The total length of the tables will be two more than three times the width. Find the length and width of the serving table so Sophia can purchase the correct size tablecloth . Round answer to the nearest tenth.

5. The couple took a small airplane for a quick flight up to the wine country for a romantic dinner and then returned home. The plane flew a total of 5 hours and each way the trip was 360 miles. If the plane was flying at 150 mph, what was the speed of the wind that affected the plane?

1. $y={x}^{2}+16x+64$

2. $y=-3{x}^{2}+12x-10$

3. A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation $A(x)=x\left(90-\frac{x}{2}\right)$ gives the area, A, of the yard for the length, x, of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.

4. $f(x)=2{x}^{2}-4x-4$

5. $f(x)=3{x}^{2}-6x-1$

1. $2{x}^{2}+x-10>0$

2. The length of a diagonal of a rectangle is three more than the width. The length of the rectangle is three times the width. Find the length of the diagonal. (Round to the nearest tenth.)