6.4.H: Homework

College Technical Math · LibreTexts

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    From LibreTexts, under CC BY 4.0, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Practice Makes Perfect

    Solve Quadratic Equations of the form \(ax^2=k\) Using the Square Root Property

    In the following exercises, solve the following quadratic equations.

    Example 1

    \(a^2=49\)

    Answer

    \(a=\pm7\)

    Example 2

    \(r^2−24=0\)

    Answer

    \(r=\pm2\sqrt{6}\)

    Example 3

    \(4m^2=36\)

    Answer

    \(m=\pm3\)

    Example 4

    \(x^2+20=0\)

    Answer

    no real solution

    Solve Quadratic Equations of the Form \(a(x−h)^2=k\) Using the Square Root Property

    In the following exercises, solve the following quadratic equations.

    Example 5

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

    Answer

    x=1, x=−5

    Example 6

    \((m−6)^2=20\)

    Answer

    \(m=6\pm2\sqrt{5}\)

    Example 7

    \((5c+1)^2=−27\)

    Answer

    no real solution

    Application

    Example 8

    Paola has enough mulch to cover 48 square feet. She wants to use it to make three square vegetable gardens of equal sizes. Solve the equation \(3s^2=48\) to find s, the length of each garden side.

    Answer

    4 feet

    Complete the Square of a Binomial Expression

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

    Example 9

    \(a^2+10a\)

    Answer

    \((a+5)^2\)

    Example 10

    \(m^2+18m\)

    Answer

    \((m+9)^2\)

    Example 11

    \(x^2−9x\)

    Answer

    \((x−\frac{9}{2})^2\)

    Solve Quadratic Equations of the Form \(x^2+bx+c=0\) by Completing the Square

    In the following exercises, solve by completing the square.

    Example 12

    \(v^2+6v=40\)

    Answer

    \(v=−10\), \(v=4\)

    Example 13

    \(u^2+2u=3\)

    Answer

    \(u=−3\), \(u=1\)

    Solve Quadratic Equations of the Form \(ax^2+bx+c=0\) by Completing the Square

    In the following exercises, solve by completing the square.

    Example 14

    \(3m^2+30m−27=6\)

    Answer

    \(m=−11\), \(m=1\)

    Example 15

    \(2c^2+c=6\)

    Answer

    \(c=−2\), \(c=\frac{3}{2}\)

    Solve Quadratic Equations Using the Quadratic Formula

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

    Example 16

    \(4m^2+m−3=0\)

    Answer

    \(m=−1\), \(m=\frac{3}{4}\)

    Example 17

    \(2p^2−7p+3=0\)

    Answer

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

    Example 18

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

    Answer

    \(p=−4\), \(p=−3\)

    Example 19

    \(r^2−8r−33=0\)

    Answer

    \(r=−3\), \(r=11\)

    Example 20

    \(3u^2+7u−2=0\)

    Answer

    \(u=\frac{−7\pm\sqrt{73}}{6}\)

    Example 21

    \(2a^2−6a+3=0\)

    Answer

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

    Example 22

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

    Answer

    no real solution

    Example 23

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

    Answer

    \(m=−1\), \(m=\frac{3}{4}\)

    Example 24

    \(2c^2−3=0\)

    Answer

    \(c=\pm\frac{\sqrt{6}}{2}\)

    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 25
    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

    Application

    Solve

    Example 26

    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

    This page titled 6.4.H: Homework was last modified on Mon, 06 Jul 2026 03:20:49 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Kiersten Purves.