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4.5E: Exercises

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    Practice Makes Perfect

     

     

    Solve Quadratic Equations of the Form \(ax^2 + bx + c = 0\) Using Factoring Methods
    1. \(x^2 - x - 6 = 0\)
    2. \(3x^2 + 7x + 4 = 0\)
    3. \(m^2 +5x + 4 = 0\)
    4. \(-2a^2 + 6a - 4 = 0\)
    5. \(p^2 + 11p + 30 = 0\)
    6. \(y^2 -18y + 45 = 0\)

    Some problems need to be rearranged a bit to fit the format \(ax^2 + bx + c = 0\) before factoring to solve:

    1. \(a^2 + 25 + 50 = -50\)
    2. \(p^2 + 3p + 67 = -16p + 19\)
    3. \(-21b = 5b^2 + 4\)
    4. \(60y^2 + 290y = 50\)
    Answer
    1. \(x = 3, x = -2\)
    2. \(x = -\frac{1}{3}, x = -1\)
    3. \(m = -1, m = -4\)
    4. \(a = 1, a = 2\)
    5. \(p = -5, p = -6\)
    6. \( y = 3, y = 15\)
    7. \(a = -5\)
    8. \(p = -3, p = -16\)
    9. \(b = -\frac{1}{5}, b = -4\)
    10. \(y = \frac{1}{6}, y = -5\)

     

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

    In the following exercises, solve each equation.

    1. \(a^{2}=49\)
    2. \(r^{2}-24=0\)
    3. \(u^{2}-300=0\)
    4. \(4 m^{2}=36\)
    5. \(\frac{4}{3} x^{2}=48\)
    6. \(x^{2}+25=0\)
    7. \(x^{2}+63=0\)
    8. \(\frac{4}{3} x^{2}+2=110\)
    9. \(\frac{2}{5} a^{2}+3=11\)
    10. \(7 p^{2}+10=26\)
    11. \(5 y^{2}-7=25\)
    Answer
    1. \(a=\pm 7\)
    2. \(r=\pm 2 \sqrt{6}\)
    3. \(u=\pm 10 \sqrt{3}\)
    4. \(m=\pm 3\)
    5. \(x=\pm 6\)
    6. \(x=\pm 5 i\)
    7. \(x=\pm 3 \sqrt{7} i\)
    8. \(x=\pm 9\)
    9. \(a=\pm 2 \sqrt{5}\)
    10. \(p=\pm \frac{4}{\sqrt{7}}\)
    11. \(y=\pm \frac{4 \sqrt{2}}{\sqrt{5}}\)
    Solve Quadratic Equations Using the Quadratic Formula

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

    1. \(4 m^{2}+m-3=0\)
    2. \(2 p^{2}-7 p+3=0\)
    3. \(p^{2}+7 p+12=0\)
    4. \(r^{2}-8 r=33\)
    5. \(3 u^{2}+7 u-2=0\)
    6. \(2 a^{2}-6 a+3=0\)
    7. \(x^{2}+8 x-4=0\)
    8. \(3 y^{2}+5 y-2=0\)
    9. \(2 x^{2}+3 x+3=0\)
    10. \(8 x^{2}-6 x+2=0\)
    11. \((v+1)(v-5)-4=0\)
    12. \(\dfrac{1}{4} m^{2}+\dfrac{1}{12} m=\dfrac{1}{3}\)
    13. \(16 c^{2}+24 c+9=0\)
    14. \(25 q^{2}+30 q+9=0\)
    Answer
    1. \(m=-1, m=\dfrac{3}{4}\)
    2. \(p=\dfrac{1}{2}, p=3\)
    3. \(p=-4, p=-3\)
    4. \(r=-3, r=11\)
    5. \(u=\dfrac{-7 \pm \sqrt{73}}{6}\)
    6. \(a=\dfrac{3 \pm \sqrt{3}}{2}\)
    7. \(x=-4 \pm 2 \sqrt{5}\)
    8. \(y=\dfrac{1}{3}, y=-2\)
    9. \(x=-\dfrac{3}{4} \pm \dfrac{\sqrt{15}}{4} i\)
    10. \(x=\dfrac{3}{8} \pm \dfrac{\sqrt{7}}{8} i\)
    11. \(v=2 \pm \sqrt{13}\)
    12. \(m=1, m=\dfrac{-4}{3}\)
    13. \(c=-\dfrac{3}{4}\)
    14. \(q=-\dfrac{3}{5}\)
    Use the Discriminant to Predict the Number of Real Solutions of a Quadratic Equation

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

    1. \(4 x^{2}-5 x+16=0\)
    2. \(36 y^{2}+36 y+9=0\)
    3. \(6 m^{2}+3 m-5=0\)
    Answer

    36. a. no real solutions b. \(1\) c. \(2\)

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

    1. \(r^{2}+12 r+36=0\)
    2. \(8 t^{2}-11 t+5=0\)
    3. \(3 v^{2}-5 v-1=0\)
    Answer

    37. a. \(1\) b. no real solutions c. \(2\)

    This page titled 4.5E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Stanislav A. Trunov and Elizabeth J. Hale via source content that was edited to the style and standards of the LibreTexts platform.