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Mathematics LibreTexts

9.3E: Exercises

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    Practice Makes Perfect

    ExerciseS 1 - 32: 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. \(4 n^{2}-9 n+5=0\)

    3. \(2 p^{2}-7 p+3=0\)

    4. \(3 q^{2}+8 q-3=0\)

    5. \(p^{2}+7 p+12=0\)

    6. \(q^{2}+3 q-18=0\)

    7. \(r^{2}-8 r=33\)

    8. \(t^{2}+13 t=-40\)

    9. \(3 u^{2}+7 u-2=0\)

    10. \(2 p^{2}+8 p+5=0\)

    11. \(2 a^{2}-6 a+3=0\)

    12. \(5 b^{2}+2 b-4=0\)

    13. \(x^{2}+8 x-4=0\)

    14. \(y^{2}+4 y-4=0\)

    15. \(3 y^{2}+5 y-2=0\)

    16. \(6 x^{2}+2 x-20=0\)

    17. \(2 x^{2}+3 x+3=0\)

    18. \(2 x^{2}-x+1=0\)

    19. \(8 x^{2}-6 x+2=0\)

    20. \(8 x^{2}-4 x+1=0\)

    21. \((v+1)(v-5)-4=0\)

    22. \((x+1)(x-3)=2\)

    23. \((y+4)(y-7)=18\)

    24. \((x+2)(x+6)=21\)

    25. \(\dfrac{1}{3} m^{2}+\dfrac{1}{12} m=\dfrac{1}{4}\)

    26. \(\dfrac{1}{3} n^{2}+n=-\dfrac{1}{2}\)

    27. \(\dfrac{3}{4} b^{2}+\dfrac{1}{2} b=\dfrac{3}{8}\)

    28. \(\dfrac{1}{9} c^{2}+\dfrac{2}{3} c=3\)

    29. \(16 c^{2}+24 c+9=0\)

    30. \(25 d^{2}-60 d+36=0\)

    31. \(25 q^{2}+30 q+9=0\)

    32. \(16 y^{2}+8 y+1=0\)

    Answer

    1. \(m=-1, m=\dfrac{3}{4}\)

    3. \(p=\dfrac{1}{3}, p=2\)

    5. \(p=-4, p=-3\)

    7. \(r=-3, r=11\)

    9. \(u=\dfrac{-7 \pm \sqrt{73}}{6}\)

    11. \(a=\dfrac{3 \pm \sqrt{3}}{2}\)

    13. \(x=-4 \pm 2 \sqrt{5}\)

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

    17. \(x=-\dfrac{3}{4} \pm \dfrac{\sqrt{15}}{4} i\)

    19. \(x=\dfrac{3}{8} \pm \dfrac{\sqrt{7}}{8} i\)

    21. \(v=2 \pm 2 \sqrt{2}\)

    23. \(y=-4, y=7\)

    25. \(m=-1, m=\dfrac{3}{4}\)

    27. \(b=\dfrac{-2 \pm \sqrt{11}}{6}\)

    29. \(c=-\dfrac{3}{4}\)

    31. \(q=-\dfrac{3}{5}\)

    ExerciseS 33 - 36 Use the Discriminant to Predict the Number of Real Solutions of a Quadratic Equation

    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. \(6 m^{2}+3 m-5=0\)
      1. \(9 v^{2}-15 v+25=0\)
      2. \(100 w^{2}+60 w+9=0\)
      3. \(5 c^{2}+7 c-10=0\)
      1. \(r^{2}+12 r+36=0\)
      2. \(8 t^{2}-11 t+5=0\)
      3. \(3 v^{2}-5 v-1=0\)
      1. \(25 p^{2}+10 p+1=0\)
      2. \(7 q^{2}-3 q-6=0\)
      3. \(7 y^{2}+2 y+8=0\)
    Answer

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

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

    ExerciseS 37 - 40: 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.

      1. \(x^{2}-5 x-24=0\)
      2. \((y+5)^{2}=12\)
      3. \(14 m^{2}+3 m=11\)
      1. \((8 v+3)^{2}=81\)
      2. \(w^{2}-9 w-22=0\)
      3. \(4 n^{2}-10=6\)
      1. \(6 a^{2}+14=20\)
      2. \(\left(x-\dfrac{1}{4}\right)^{2}=\dfrac{5}{16}\)
      3. \(y^{2}-2 y=8\)
      1. \(8 b^{2}+15 b=4\)
      2. \(\dfrac{5}{9} v^{2}-\dfrac{2}{3} v=1\)
      3. \(\left(w+\dfrac{4}{3}\right)^{2}=\dfrac{2}{9}\)
    Answer

    37. a. Factor b. Square Root c. Quadratic Formula

    39. a. Quadratic Formula b. Square Root c. Factor

    ExerciseS 41 - 42: Writing Exercises

    1. Solve the equation \(x^{2}+10 x=120\)
      1. by completing the square
      2. using the Quadratic Formula
      3. Which method do you prefer? Why?
    2. Solve the equation \(12 y^{2}+23 y=24\)
      1. by completing the square
      2. using the Quadratic Formula
      3. Which method do you prefer? Why?
    Answer

    41. Answers will vary

    Self Check

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

    This table provides a checklist to evaluate mastery of the objectives of this section. Choose how would you respond to the statement “I can solve quadratic equations using the quadratic formula.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can use the discriminant to predict the number of solutions of a quadratic equation.” “Confidently,” “with some help,” or “No, I don’t get it.” Choose how would you respond to the statement “I can identify the most appropriate method to use to solve a quadratic equation.” “Confidently,” “with some help,” or “No, I don’t get it.”
    Figure 9.3.87

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