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2.EE: Exercises for Quadratic Equations

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    Solving Quadratic Equations Using the Square Root Property

    Exercise A

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

    1. \(y^{2}=144\)
    2. \(n^{2}-80=0\)
    3. \(4 a^{2}=100\)
    4. \(2 b^{2}=72\)
    5. \(r^{2}+32=0\)
    6. \(t^{2}+18=0\)
    7. \(\frac{2}{3} w^{2}-20=30\)
    8. \(5 c^{2}+3=19\)
    Answer

    1. \(y=\pm 12\)

    3. \(a=\pm 5\)

    5. \(r=\pm 4 \sqrt{2} i\)

    7. \(w=\pm 5 \sqrt{3}\)

    Exercise B

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

    1. \((p-5)^{2}+3=19\)
    2. \((u+1)^{2}=45\)
    3. \(\left(x-\frac{1}{4}\right)^{2}=\frac{3}{16}\)
    4. \(\left(y-\frac{2}{3}\right)^{2}=\frac{2}{9}\)
    5. \((n-4)^{2}-50=150\)
    6. \((4 c-1)^{2}=-18\)
    7. \(n^{2}+10 n+25=12\)
    8. \(64 a^{2}+48 a+9=81\)
    Answer

    1. \(p=-1,9\)

    3. \(x=\frac{1}{4} \pm \frac{\sqrt{3}}{4}\)

    5. \(n=4 \pm 10 \sqrt{2}\)

    7. \(n=-5 \pm 2 \sqrt{3}\)

    Solving Quadratic Equations by Completing the Square

    Exercise A

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

    1. \(x^{2}+22 x\)
    2. \(m^{2}-8 m\)
    3. \(a^{2}-3 a\)
    4. \(b^{2}+13 b\)
    Answer

    1. \((x+11)^{2}\)

    3. \(\left(a-\frac{3}{2}\right)^{2}\)

    Exercise B

    In the following exercises, solve by completing the square.

    1. \(d^{2}+14 d=-13\)
    2. \(y^{2}-6 y=36\)
    3. \(m^{2}+6 m=-109\)
    4. \(t^{2}-12 t=-40\)
    5. \(v^{2}-14 v=-31\)
    6. \(w^{2}-20 w=100\)
    7. \(m^{2}+10 m-4=-13\)
    8. \(n^{2}-6 n+11=34\)
    9. \(a^{2}=3 a+8\)
    10. \(b^{2}=11 b-5\)
    11. \((u+8)(u+4)=14\)
    12. \((z-10)(z+2)=28\)
    Answer

    1. \(d=-13,-1\)

    3. \(m=-3 \pm 10 i\)

    5. \(v=7 \pm 3 \sqrt{2}\)

    7. \(m=-9,-1\)

    9. \(a=\frac{3}{2} \pm \frac{\sqrt{41}}{2}\)

    11. \(u=-6 \pm 2 \sqrt{2}\)

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

    Exercise A

    In the following exercises, solve by completing the square.

    1. \(3 p^{2}-18 p+15=15\)
    2. \(5 q^{2}+70 q+20=0\)
    3. \(4 y^{2}-6 y=4\)
    4. \(2 x^{2}+2 x=4\)
    5. \(3 c^{2}+2 c=9\)
    6. \(4 d^{2}-2 d=8\)
    7. \(2 x^{2}+6 x=-5\)
    8. \(2 x^{2}+4 x=-5\)
    Answer

    1. \(p=0,6\)

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

    5. \(c=-\frac{1}{3} \pm \frac{2 \sqrt{7}}{3}\)

    7. \(x=\frac{3}{2} \pm \frac{1}{2} i\)

    Exercise B

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

    1. \(4 x^{2}-5 x+1=0\)
    2. \(7 y^{2}+4 y-3=0\)
    3. \(r^{2}-r-42=0\)
    4. \(t^{2}+13 t+22=0\)
    5. \(4 v^{2}+v-5=0\)
    6. \(2 w^{2}+9 w+2=0\)
    7. \(3 m^{2}+8 m+2=0\)
    8. \(5 n^{2}+2 n-1=0\)
    9. \(6 a^{2}-5 a+2=0\)
    10. \(4 b^{2}-b+8=0\)
    11. \(u(u-10)+3=0\)
    12. \(5 z(z-2)=3\)
    13. \(\frac{1}{8} p^{2}-\frac{1}{5} p=-\frac{1}{20}\)
    14. \(\frac{2}{5} q^{2}+\frac{3}{10} q=\frac{1}{10}\)
    15. \(4 c^{2}+4 c+1=0\)
    16. \(9 d^{2}-12 d=-4\)
    Answer

    1. \(x=\frac{1}{4}, 1\)

    3. \(r=-6,7\)

    5. \(v=\frac{-1 \pm \sqrt{21}}{8}\)

    7. \(m=\frac{-4 \pm \sqrt{10}}{3}\)

    9. \(a=\frac{5}{12} \pm \frac{\sqrt{23}}{12} i\)

    11. \(u=5 \pm \sqrt{21}\)

    13. \(p=\frac{4 \pm \sqrt{5}}{5}\)

    15. \(c=-\frac{1}{2}\)

    Exercise C

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

      1. \(9 x^{2}-6 x+1=0\)
      2. \(3 y^{2}-8 y+1=0\)
      3. \(7 m^{2}+12 m+4=0\)
      4. \(5 n^{2}-n+1=0\)
      1. \(5 x^{2}-7 x-8=0\)
      2. \(7 x^{2}-10 x+5=0\)
      3. \(25 x^{2}-90 x+81=0\)
      4. \(15 x^{2}-8 x+4=0\)
    Answer

    1.

    1. \(1\)
    2. \(2\)
    3. \(2\)
    4. \(2\)
    Exercise D

    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.

      1. \(16 r^{2}-8 r+1=0\)
      2. \(5 t^{2}-8 t+3=9\)
      3. \(3(c+2)^{2}=15\)
      1. \(4 d^{2}+10 d-5=21\)
      2. \(25 x^{2}-60 x+36=0\)
      3. \(6(5 v-7)^{2}=150\)
    Answer

    1.

    1. Factor
    2. Quadratic Formula
    3. Square Root

    Solving Equations in Quadratic Form

    Exercise A

    In the following exercises, solve.

    1. \(x^{4}-14 x^{2}+24=0\)
    2. \(x^{4}+4 x^{2}-32=0\)
    3. \(4 x^{4}-5 x^{2}+1=0\)
    4. \((2 y+3)^{2}+3(2 y+3)-28=0\)
    5. \(x+3 \sqrt{x}-28=0\)
    6. \(6 x+5 \sqrt{x}-6=0\)
    7. \(x^{\frac{2}{3}}-10 x^{\frac{1}{3}}+24=0\)
    8. \(x+7 x^{\frac{1}{2}}+6=0\)
    9. \(8 x^{-2}-2 x^{-1}-3=0\)
    Answer

    1. \(x=\pm \sqrt{2}, x=\pm 2 \sqrt{3}\)

    3. \(x=\pm 1, x=\pm \frac{1}{2}\)

    5. \(x=16\)

    7. \(x=64, x=216\)

    9. \(x=-2, x=\frac{4}{3}\)

    This page titled 2.EE: Exercises for Quadratic Equations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Katherine Skelton.