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10.3E: Exercises

  • Page ID
    30285
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    Practice Makes Perfect

    Solve Quadratic Equations Using the Quadratic Formula

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

    Example \(\PageIndex{31}\)

    \(4m^2+m−3=0\)

    Answer

    \(m=−1\), \(m=\frac{3}{4}\)

    Example \(\PageIndex{32}\)

    \(4n^2−9n+5=0\)

    Example \(\PageIndex{33}\)

    \(2p^2−7p+3=0\)

    Answer

    \(p=\frac{1}{2}\), \(p=3\)

    Example \(\PageIndex{34}\)

    \(3q^2+8q−3=0\)

    Example \(\PageIndex{35}\)

    \(p^2+7p+12=0\)

    Answer

    \(p=−4\), \(p=−3\)

    Example \(\PageIndex{36}\)

    \(q^2+3q−18=0\)

    Example \(\PageIndex{37}\)

    \(r^2−8r−33=0\)

    Answer

    \(r=−3\), \(r=11\)

    Example \(\PageIndex{38}\)

    \(t^2+13t+40=0\)

    Example \(\PageIndex{39}\)

    \(3u^2+7u−2=0\)

    Answer

    \(u=\frac{−7\pm\sqrt{73}}{6}\)

    Example \(\PageIndex{40}\)

    \(6z^2−9z+1=0\)

    Example \(\PageIndex{41}\)

    \(2a^2−6a+3=0\)

    Answer

    \(a=\frac{3\pm\sqrt{3}}{2}\)

    Example \(\PageIndex{42}\)

    \(5b^2+2b−4=0\)

    Example \(\PageIndex{43}\)

    \(2x^2+3x+9=0\)

    Answer

    no real solution

    Example \(\PageIndex{44}\)

    \(6y^2−5y+2=0\)​​​​​​

    Example \(\PageIndex{45}\)

    \(v(v+5)−10=0\)

    Answer

    \(v=\frac{−5\pm\sqrt{65}}{2}\)

    Example \(\PageIndex{46}\)

    \(3w(w−2)−8=0\)​​​​​​

    Example \(\PageIndex{47}\)

    \(\frac{1}{3}m^2+\frac{1}{12}m=\frac{1}{4}\)

    Answer

    \(m=−1\), \(m=\frac{3}{4}\)

    Example \(\PageIndex{48}\)

    \(\frac{1}{3}n^2+n=−\frac{1}{2}\)

    Example \(\PageIndex{49}\)

    \(16c^2+24c+9=0\)

    Answer

    \(c=−\frac{3}{4}\)

    Example \(\PageIndex{50}\)

    \(25d^2−60d+36=0\)

    Example \(\PageIndex{51}\)

    5m^2+2m−7=0

    Answer

    \(m=−\frac{7}{5}\), \(m=1\)

    Example \(\PageIndex{52}\)

    \(8n^2−3n+3=0\)​​​​​​

    Example \(\PageIndex{53}\)

    \(p^2−6p−27=0\)

    Answer

    \(p=−3\), \(p=9\)

    Example \(\PageIndex{54}\)

    \(25q^2+30q+9=0\)

    Example \(\PageIndex{55}\)

    \(4r^2+3r−5=0\)

    Answer

    \(r=\frac{−3\pm\sqrt{89}}{8}\)​​​​​

    Example \(\PageIndex{56}\)

    \(3t(t−2)=2\)​​​​​​​

    Example \(\PageIndex{57}\)

    \(2a^2+12a+5=0\)

    Answer

    \(a=\frac{−6\pm\sqrt{26}}{2}\)​​​​​​​

    Example \(\PageIndex{58}\)

    \(4d^2−7d+2=0\)​​​​​​​

    Example \(\PageIndex{59}\)

    \(\frac{3}{4}b^2+\frac{1}{2}b=\frac{3}{8}\)

    Answer

    \(b=\frac{−2\pm\sqrt{11}}{6}\)​​​​​​​

    Example \(\PageIndex{60}\)

    \(\frac{1}{9}c^2+\frac{2}{3}c=3\)​​​​​​​

    Example \(\PageIndex{61}\)

    \(2x^2+12x−3=0\)

    Answer

    \(x=\frac{−6\pm\sqrt{42}}{4}\)​​​​​​​

    Example \(\PageIndex{62}\)

    \(16y^2+8y+1=0\)

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

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

    Example \(\PageIndex{63}\)
    1. \(4x^2−5x+16=0\)
    2. \(36y^2+36y+9=0\)
    3. \(6m^2+3m−5=0\)
    4. \(18n^2−7n+3=0\)
    Answer
    1. no real solutions
    2. 1
    3. 2
    4. no real solutions
    Example \(\PageIndex{64}\)
    1. \(9v^2−15v+25=0\)
    2. \(100w^2+60w+9=0\)
    3. \(5c^2+7c−10=0\)
    4. \(15d^2−4d+8=0\)
    Example \(\PageIndex{65}\)
    1. \(r^2+12r+36=0\)
    2. \(8t^2−11t+5=0\)
    3. \(4u^2−12u+9=0\)
    4. \(3v^2−5v−1=0\)
    Answer
    1. 1
    2. no real solutions
    3. 1
    4. 2
    Example \(\PageIndex{66}\)
    1. \(25p^2+10p+1=0\)
    2. \(7q^2−3q−6=0\)
    3. \(7y^2+2y+8=0\)
    4. \(25z^2−60z+36=0\)

    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.

    Example \(\PageIndex{67}\)
    1. \(x^2−5x−24=0\)
    2. \((y+5)^2=12\)
    3. \(14m^2+3m=11\)
    Answer
    1. factor
    2. square root
    3. Quadratic Formula
    Example \(\PageIndex{68}\)
    1. \((8v+3)^2=81\)
    2. \(w^2−9w−22=0\)
    3. \(4n^2−10=6\)​​​​​​​
    Example \(\PageIndex{69}\)
    1. \(6a^2+14=20\)
    2. \((x−\frac{1}{4})^2=\frac{5}{16}\)
    3. \(y^2−2y=8\)
    Answer
    1. factor
    2. square root
    3. factor​​​​​​​
    Example \(\PageIndex{70}\)
    1. \(8b^2+15b=4\)
    2. \(\frac{5}{9}v^2−\frac{2}{3}v=1\)
    3. \((w+\frac{4}{3})^2=\frac{2}{9}\)

    Everyday Math

    Example \(\PageIndex{71}\)

    A flare is fired straight up from a ship at sea. Solve the equation \(16(t^2−13t+40)=0\) for t, the number of seconds it will take for the flare to be at an altitude of 640 feet.

    Answer

    5 seconds, 8 seconds

    Example \(\PageIndex{72}\)

    An architect is designing a hotel lobby. She wants to have a triangular window looking out to an atrium, with the width of the window 6 feet more than the height. Due to energy restrictions, the area of the window must be 140 square feet. Solve the equation \(\frac{1}{2}h^2+3h=140\) for h, the height of the window.

    Writing Exercises

    Example \(\PageIndex{73}\)

    Solve the equation \(x^2+10x=200\)

    1. by completing the square
    2. using the Quadratic Formula
    3. Which method do you prefer? Why?
    Answer
    1. −20, 10
    2. −20, 10
    3. answers will vary
    Example \(\PageIndex{74}\)

    Solve the equation \(12y^2+23y=24\)

    1. by completing the square
    2. using the Quadratic Formula
    3. Which method do you prefer? Why?

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has four rows and four columns. The first row is a header row and it labels each column. The first column is labeled "I can …", the second "Confidently", the third “With some help” and the last "No–I don’t get it". In the “I can…” column the next row reads “solve quadratic equations using the quadratic formula.” The next row reads “use the discriminant to predict the number of solutions of a quadratic equation.” and the last row reads “identify the most appropriate method to use to solve a quadratic equation.” The remaining columns are blank.

    ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?


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