Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

9.E: Review Exercises and Sample Exam

  • Page ID
    21627
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Review Exercises

    Exercise \(\PageIndex{1}\) extracting square roots

    Solve by extracting the roots.

    1. \(x^{2}−16=0\)
    2. \(y^{2}=94\)
    3. \(x^{2}−27=0\)
    4. \(x^{2}+27=0\)
    5. \(3y^{2}−25=0\)
    6. \(9x^{2}−2=0\)
    7. \((x−5)^{2}−9=0\)
    8. \((2x−1)^{2}−1=0\)
    9. \(16(x−6)^{2}−3=0\)
    10. \(2(x+3)^{2}−5=0\)
    11. \((x+3)(x−2)=x+12\)
    12. \((x+2)(5x−1)=9x−1\)
    Answer

    1. \(±16\)

    3. \(±3\sqrt{3}\) 

    5. \(±\frac{5 \sqrt{3}}{3}\) 

    7. \(2, 8\)

    9. \(6±\frac{\sqrt{3}}{4}\) 

    11. \(±3\sqrt{2}\)

    Exercise \(\PageIndex{2}\) extracting square roots

    Find a quadratic equation in standard form with the given solutions.

    1. \(\pm\sqrt{2}\)
    2. \(\pm2\sqrt{5}\)
    Answer

    1. \(x^{2}-2=0\)

    Exercise \(\PageIndex{3}\) completing the square

    Complete the square.

    1. \(x^{2}-6x+?=(x-?)^{2}\)
    2. \(x^{2}-x+?=(x-?)^{2}\)
    Answer

    1. \(x^{2}-6x+9=(x-3)^{2}\)

    Exercise \(\PageIndex{4}\) completing the square

    Solve by completing the square.

    1. \(x^{2}−12x+1=0\)
    2. \(x^{2}+8x+3=0\)
    3. \(y^{2}−4y−14=0\)
    4. \(y^{2}−2y−74=0\)
    5. \(x^{2}+5x−1=0\)
    6. \(x^{2}−7x−2=0\)
    7. \(2x^{2}+x−3=0\)
    8. \(5x^{2}+9x−2=0\)
    9. \(2x^{2}−16x+5=0\)
    10. \(3x^{2}−6x+1=0\)
    11. \(2y^{2}+10y+1=0\)
    12. \(5y^{2}+y−3=0\)
    13. \(x(x+9)=5x+8\)
    14. \((2x+5)(x+2)=8x+7\)
    Answer

    1. \(6±\sqrt{35}\)

    3. \(2±3\sqrt{2}\)

    5. \(\frac{-5±\sqrt{29}}{2}\)

    7. \(\frac{−3}{2}, 1\)

    9. \(\frac{8±3\sqrt{6}}{2}\)

    11. \(\frac{-5±\sqrt{23}}{2}\)

    13. \(−2±2\sqrt{3}\)

    Exercise \(\PageIndex{5}\) quadratic formula

    Identify the coefficients a, b, and c used in the quadratic formula. Do not solve.

    1. \(x^{2}−x+4=0\)
    2. \(−x^{2}+5x−14=0\)
    3. \(x^{2}−5=0\)
    4. \(6x^{2}+x=0\)
    Answer

    1. \(a=1, b=−1,\) and \(c=4\)

    3. \(a=1, b=0,\) and \(c=−5\)

    Exercise \(\PageIndex{6}\) quadratic formula

    Use the quadratic formula to solve the following.

    1. \(x^{2}−6x+6=0\)
    2. \(x^{2}+10x+23=0\)
    3. \(3y^{2}−y−1=0\)
    4. \(2y^{2}−3y+5=0\)
    5. \(5x^{2}−36=0\)
    6. \(7x^{2}+2x=0\)
    7. \(−x^{2}+5x+1=0\)
    8. \(−4x^{2}−2x+1=0\)
    9. \(t^{2}−12t−288=0\)
    10. \(t^{2}−44t+484=0\)
    11. \((x−3)^{2}−2x=47\)
    12. \(9x(x+1)−5=3x\)
    Answer

    1. \(3±\sqrt{3}\)

    3. \(\frac{1±\sqrt{13}}{6}\)

    5. \(±\frac{6\sqrt{5}}{5}\)

    7. \(\frac{5±\sqrt{29}}{2}\) 

    9. \(−12, 24 \)

    11. \(4±3\sqrt{6}\)

    Exercise \(\PageIndex{7}\) Guidelines for Solving Quadratic Equations and Applications

    Use the discriminant to determine the number and type of solutions.

    1. \(−x^{2}+5x+1=0\)
    2. \(−x^{2}+x−1=0\)
    3. \(4x^{2}−4x+1=0\)
    4. \(9x^{2}−4=0\)
    Answer

    1. Two real solutions

    3. One real solution

    Exercise \(\PageIndex{8}\) Guidelines for Solving Quadratic Equations and Applications

    Solve using any method.

    1. \(x^{2}+4x−60=0\)
    2. \(9x^{2}+7x=0\)
    3. \(25t^{2}−1=0\)
    4. \(t^{2}+16=0\)
    5. \(x^{2}−x−3=0\)
    6. \(9x^{2}+12x+1=0\)
    7. \(4(x−1)^{2}−27=0\)
    8. \((3x+5)^{2}−4=0\)
    9. \((x−2)(x+3)=6\)
    10. \(x(x−5)=12\)
    11. \((x+1)(x−8)+28=3x\)
    12. \((9x−2)(x+4)=28x−9\)
    Answer

    1. \(−10, 6\)

    3. \(±\frac{1}{5}\)

    5. \(\frac{1±\sqrt{13}}{2}\)

    7. \(1 ± \frac{3 \sqrt{3}}{2}\)

    9. \(−4, 3\)

    11. \(5±\sqrt{5}\)

    Exercise \(\PageIndex{9}\) Guidelines for Solving Quadratic Equations and Applications

    Set up an algebraic equation and use it to solve the following.

    1. The length of a rectangle is 2 inches less than twice the width. If the area measures 25 square inches, then find the dimensions of the rectangle. Round off to the nearest hundredth.
    2. An 18-foot ladder leaning against a building reaches a height of 17 feet. How far is the base of the ladder from the wall? Round to the nearest tenth of a foot.
    3. The value in dollars of a new car is modeled by the function \(V(t)=125t^{2}−3,000t+22,000\), where t represents the number of years since it was purchased. Determine the age of the car when its value is $22,000.
    4. The height in feet reached by a baseball tossed upward at a speed of 48 feet/second from the ground is given by the function \(h(t)=−16t^{2}+48t\), where t represents time in seconds. At what time will the baseball reach a height of 16 feet?
    Answer

    1. Length: 6.14 inches; width: 4.07 inches

    3. It is worth $22,000 new and when it is 24 years old.

    Exercise \(\PageIndex{10}\) graphing parabolas

    Determine the x- and y-intercepts.

    1. \(y=2x^{2}+5x−3\)
    2. \(y=x^{2}−12\)
    3. \(y=5x^{2}−x+2\)
    4. \(y=−x^{2}+10x−25\)
    Answer

    1. x-intercepts: \((−3, 0), (\frac{1}{2}, 0)\); y-intercept: \((0, −3)\)

    3. x-intercepts: none; y-intercept: \((0, 2)\)

    Exercise \(\PageIndex{11}\) graphing parabolas

    Find the vertex and the line of symmetry.

    1. \(y=x^{2}−6x+1\)
    2. \(y=−x^{2}+8x−1\)
    3. \(y=x^{2}+3x−1\)
    4. \(y=9x^{2}−1\)
    Answer

    1. Vertex: \((3, −8)\); line of symmetry: \(x=3\)

    3. Vertex: \((−\frac{3}{2}, −\frac{13}{4})\); line of symmetry: \(x=−32\)

    Exercise \(\PageIndex{12}\) graphing parabolas

    Graph. Find the vertex and the y-intercept. In addition, find the x-intercepts if they exist.

    1. \(y=x^{2}+8x+12\)
    2. \(y=−x^{2}−6x+7\)
    3. \(y=−2x^{2}−4\)
    4. \(y=x^{2}+4x\)
    5. \(y=4x^{2}−4x+1\)
    6. \(y=−2x^{2}\)
    7. \(y=−2x^{2}+8x−7\)
    8. \(y=3x^{2}−1\)
    Answer

    1. 

    Screenshot (276).png
    Figure 9.E.1

    3. 

    Screenshot (277).png
    Figure 9.E.2

    5. 

    Screenshot (278).png
    Figure 9.E.3

    7.

    Screenshot (279).png
    Figure 9.E.4

    Exercise \(\PageIndex{13}\) graphing parabolas

    Determine the maximum or minimum y-value.

    1. \(y=x^{2}−10x+1\)
    2. \(y=−x^{2}+12x−1\)
    3. \(y=−5x^{2}+6x\)
    4. \(y=2x^{2}−x−1\)
    5. The value in dollars of a new car is modeled by the function \(V(t)=125t^{2}−3,000t+22,000\), where t represents the number of years since it was purchased. Determine the age of the car when its value is at a minimum. 
    6. The height in feet reached by a baseball tossed upward at a speed of 48 feet/second from the ground is given by the function \(h(t)=−16t^{2}+48t\), where t represents time in seconds. What is the maximum height of the baseball?
    Answer

    1. Minimum: \(y = −24\)

    3. Maximum: \(y = \frac{9}{5}\)

    5. The car will have a minimum value 12 years after it is purchased.

    Exercise \(\PageIndex{14}\) introduction to complex numbers and complex solutions

    Rewrite in terms of i.

    1. \(\sqrt{−36}\)
    2. \(\sqrt{−40}\)
    3. \(\sqrt{−\frac{8}{25}}\)
    4. -\(\sqrt{−19}\)
    Answer

    1. 6i

    3. \(\frac{2 \sqrt{2} i}{5}\)

    Exercise \(\PageIndex{15}\) introduction to complex numbers and complex solutions

    Perform the operations.

    1. \((2−5i)+(3+4i)\)
    2. \((6−7i)−(12−3i)\)
    3. \((2−3i)(5+i)\)
    4. \(4−i^{2}−3i\)
    Answer

    1. \(5−i\)

    3. \(13−13i\)

    Exercise \(\PageIndex{16}\) introduction to complex numbers and complex solutions

    Solve.

    1. \(9x^{2}+25=0\)
    2. \(3x^{2}+1=0\)
    3. \(y^{2}−y+5=0\)
    4. \(y^{2}+2y+4\)
    5. \(4x(x+2)+5=8x\)
    6. \(2(x+2)(x+3)=3(x^{2}+13)\)
    Answer

    1. \(\pm\frac{3}{3}\)

    3. \(\frac{1}{2}\pm i \frac{\sqrt{19}}{2}\)

    5. \(\pm i \frac{\sqrt{5}}{2}\)

    Sample Exam

    Exercise \(\PageIndex{17}\)

    Solve by extracting the roots.

    1. \(4x^{2}−9=0\)
    2. \((4x+1)^{2}−5=0\)
    Answer

    1. \(\pm\frac{3}{2}\)

    Exercise \(\PageIndex{18}\)

    Solve by completing the square.

    1. \(x^{2}+10x+19=0\)
    2. \(x^{2}−x−1=0\)
    Answer

    1. \(-5\pm\sqrt{6}\)

    Exercise \(\PageIndex{19}\)

    Solve using the quadratic formula.

    1. \(−2x^{2}+x+3=0\)
    2. \(x^{2}+6x−31=0\)
    Answer

    1. \(-1, \frac{3}{2}\)

    Exercise \(\PageIndex{20}\)

    Solve using any method.

    1. \((5x+1)(x+1)=1\)
    2. \((x+5)(x−5)=65\)
    3. \(x(x+3)=−2\)
    4. \(2(x−2)^{2}−6=3x^{2}\)
    Answer

    1. \(-\frac{6}{5}, 0\)

    3. \(-2, -1\)

    Exercise \(\PageIndex{21}\)

    Set up an algebraic equation and solve.

    1. The length of a rectangle is twice its width. If the diagonal measures \(6\sqrt{5}\) centimeters, then find the dimensions of the rectangle.
    2. The height in feet reached by a model rocket launched from a platform is given by the function \(h(t)=−16t^{2}+256t+3\), where t represents time in seconds after launch. At what time will the rocket reach 451 feet?
    Answer

    1. Length: 12 centimeters; width: 6 centimeters

    Exercise \(\PageIndex{22}\)

    Graph. Find the vertex and the y-intercept. In addition, find the x-intercepts if they exist.

    1. \(y=2x^{2}−4x−6\)
    2. \(y=−x^{2}+4x−4\)
    3. \(y=4x^{2}−9\)
    4. \(y=x^{2}+2x−1\)
    5. Determine the maximum or minimum y-value: \(y=−3x^{2}+12x−15\).
    6. Determine the x- and y-intercepts: \(y=x^{2}+x+4\).
    7. Determine the domain and range: \(y=25x^{2}−10x+1\).
    8. The height in feet reached by a model rocket launched from a platform is given by the function \(h(t)=−16t^{2}+256t+3\), where t represents time in seconds after launch. What is the maximum height attained by the rocket. 
    9. A bicycle manufacturing company has determined that the weekly revenue in dollars can be modeled by the formula \(R=200n−n^{2}\), where n represents the number of bicycles produced and sold. How many bicycles does the company have to produce and sell in order to maximize revenue?
    10. Rewrite in terms of i: \(\sqrt{−60}\).
    11. Divide: \(\frac{4−2i}{4+2i}\).
    Answer

    1. 

    Screenshot (280).png
    Figure 9.E.5

    3. 

    Screenshot (281).png
    Figure 9.E.6

    5. Maximum: \(y = −3\)

    7. Domain: R; range: \([0,∞)\)

    9. To maximize revenue, the company needs to produce and sell 100 bicycles a week. 

    11. \(\frac{3}{5}−i\frac{4}{5}\)

    Exercise \(\PageIndex{23}\)

    Solve.

    1. \(25x^{2}+3=0\)
    2. \(−2x^{2}+5x−1=0\)
    Answer

    2. \(\frac{5\pm\sqrt{17}}{4}\)