Exercises - Solving Equations

Video examples of using various techniques to solve equations can be found here.
  1. Solve the following equations involving only a single occurrence of a variable:

    1. $\displaystyle{3x-7=0}$   $\ans{\displaystyle{x=\frac{7}{3}}}$

    2. $\displaystyle{\frac{5x^3+2}{7}=6}$   $\ans{\displaystyle{x=2}}$

    3. $\displaystyle{\frac{1}{3-\frac{1}{x}}=\frac{4}{11}}$   $\ans{\displaystyle{x=4}}$

    4. $\displaystyle{(\sqrt[3]{x}+5)^{-1}=2}$   $\ans{\displaystyle{x = -\frac{729}{8}}}$

    5. $\displaystyle{2x^4-8=0}$   $\ans{\displaystyle{x=\pm \sqrt{2}}}$

  2. Solve by completing the square, and then check your answer by using the quadratic formula:

    1. $\displaystyle{3x^2 - 4x - 2 = 0}$   $\ans{\displaystyle{x = \frac{2 \pm \sqrt{10}}{3}}}$

    2. $\displaystyle{z^2 + 8z - 3 = 0}$   $\ans{\displaystyle{z = -4 \pm \sqrt{19}}}$

    3. $\displaystyle{2x^2 + 5x - 1 = 0}$   $\ans{\displaystyle{\frac{-5 \pm \sqrt{33}}{4}}}$

    4. $\displaystyle{5x^2 = 13 - 2x}$   $\ans{\displaystyle{\frac{-1 \pm \sqrt{66}}{5}}}$

  3. Solve the following:

    1. $\displaystyle{3 = x(5x-1)}$   $\ans{\displaystyle{\frac{1 \pm \sqrt{61}}{10}}}$

    2. $\displaystyle{\sqrt{4-2x} - 2 = x}$   $\ans{\displaystyle{x = 0 \textrm{ only}}}$

    3. $\displaystyle{2t^{-2} + 5t^{-1} - 3 = 0}$   $\ans{\displaystyle{t = -\frac{1}{3}, 2}}$

    4. $\displaystyle{c^{2/3} - c^{1/3} - 2 =0}$   $\ans{\displaystyle{c=-1,8}}$

    5. $\displaystyle{2x^{-2} + 3x^{-1} + 1 = 0}$   $\ans{\displaystyle{x = -1,-2}}$

    6. $\displaystyle{3 \left( \frac{5}{x-1} \right)^2 + \left( \frac{5}{x-1} \right) - 2 = 0}$   $\ans{\displaystyle{x = -4, \frac{17}{2}}}$

  4. Solve for $x$:

    1. $\displaystyle{x(5x-2)=4}$   $\ans{\displaystyle{x = \frac{1 \pm \sqrt{21}}{5}}}$

    2. $\displaystyle{z = 36z^3}$   $\ans{\displaystyle{z = 0, \pm \frac{1}{6}}}$

    3. $\displaystyle{\frac{2x-1}{x^2+2x-8} - \frac{3}{2-x} = \frac{2}{x+4}}$   $\ans{\displaystyle{x = -5}}$

    4. $\displaystyle{x^4 + x^2 - 12 = 0}$   $\ans{\displaystyle{x = \pm \sqrt{3} \textrm{ only}}}$

    5. $\displaystyle{1+ \sqrt{2x+1} = x}$   $\ans{\displaystyle{x = 4 \textrm{ only}}}$

    6. $\displaystyle{2x(x+2) = 3}$   $\ans{\displaystyle{\frac{x = -2 \pm \sqrt{10}}{2}}}$

    7. $\displaystyle{2x^3 + x^2 - 2x - 1 = 0}$   $\ans{\displaystyle{ x = \pm 1, - \frac{1}{2}}}$

    8. $\displaystyle{\frac{x}{x+2} + \frac{2}{x-3} = \frac{10}{x^2 - x - 6}}$   $\ans{\displaystyle{\textrm{there are no solutions}}}$

    9. $\displaystyle{4x^{-2} - x^{-1} - 5 = 0}$   $\ans{\displaystyle{x = \frac{4}{5}, -1}}$

    10. $\displaystyle{a^{2/3} - 3a^{1/3} - 10 = 0}$   $\ans{\displaystyle{a = 125, -8}}$

    11. $\displaystyle{(a^2 - 1)^2 - (a^2 - 1) - 2 = 0}$   $\ans{\displaystyle{a = \pm \sqrt{3},0}}$

    12. $\displaystyle{\left( \frac{x+1}{x} \right)^2 + 2 \left( \frac{x+1}{x} \right) - 3 = 0}$   $\ans{\displaystyle{x = -\frac{1}{4}}}$

    13. $\displaystyle{(6x^3 + x^2 - 35x)(49 - x^4) = 0}$   $\ans{\displaystyle{x = \pm \sqrt{7^{\phantom{1}}}, 0, -\frac{5}{2}, \frac{7}{3}}}$

    14. $\displaystyle{5\sqrt[3]{x^2} - 4\sqrt[3]{x} = 1}$   $\ans{\displaystyle{x = 1, -\frac{1}{125}}}$

    15. $\displaystyle{4 \left( \frac{\sqrt[3]{x+1}}{7} + 2 \right)^5 = \frac{1}{8}}$   $\ans{\displaystyle{x = -\frac{9269}{8}}}$

    16. $\displaystyle{x^4 - 9 = -2x^2}$   $\ans{\displaystyle{x = \pm \sqrt{-1 + \sqrt{10^{\phantom{1}}}}}}$

    17. $\displaystyle{(2x^4 + 30x^3 + 150x^2 + 250x)(4x^2 - 4x + 1) = 0}$   $\ans{\displaystyle{x = 0, -5, \frac{1}{2}}}$

    18. $\displaystyle{4 \left( \sqrt{\frac{x+1}{2}} + \frac{\sqrt{2x-4}}{4} \right) = 5\sqrt{2}}$   $\ans{\displaystyle{x = 3}}$

    19. $\displaystyle{x^3 + 2x^2 - 3ax = 6a}$   $\ans{\displaystyle{x = -2, \sqrt{3a}}}$

  5. Solve:

    1. $\displaystyle{\frac{x-3}{x+3}=\frac{2}{x}}$   $\ans{\displaystyle{x=-1,6}}$

    2. $\displaystyle{2x^2-11=0}$   $\ans{\displaystyle{x=\pm \sqrt{\frac{11}{2}}}}$

    3. $\displaystyle{2x^2+5x=3}$   $\ans{\displaystyle{x=-3,\frac{1}{2}}}$

    4. $\displaystyle{x-3(2-x)=2(x+1)-2}$   $\ans{\displaystyle{x=3}}$

    5. $\displaystyle{x^4-13x^2+36=0}$   $\ans{\displaystyle{x=\pm 3, \pm 2}}$

    6. $\displaystyle{x^2+2x=5}$   $\ans{\displaystyle{x=-1 \pm \sqrt{6}}}$

    7. $\displaystyle{\frac{x}{2x+1}+\frac{1}{x+2}=2}$   $\ans{\displaystyle{x=-1}}$

    8. $\displaystyle{\sqrt{4-x^2}=1}$   $\ans{\displaystyle{x=\pm \sqrt{3}}}$

    9. $\displaystyle{18x^2=9x+20}$   $\ans{\displaystyle{x = -\frac{5}{6}, \frac{4}{3}}}$

    10. $\displaystyle{2x^2+13x-7=2x-7}$   $\ans{\displaystyle{x=-\frac{11}{2},0}}$

    11. $\displaystyle{x^3=x}$   $\ans{\displaystyle{x= \pm 1, 0}}$

    12. $\displaystyle{x=2\sqrt{x}-1}$   $\ans{\displaystyle{x=1}}$

  6. Solve:

    1. $\displaystyle{x^{2/3} + x^{1/3} - 6 = 0}$   $\ans{\displaystyle{x=-27,8}}$

    2. $\displaystyle{x^{11/3}+7x^{8/3} + 12x^{5/3} = 0}$   $\ans{\displaystyle{x=0,-3,-4}}$

    3. $\displaystyle{(x+2)^{-2} + (x+2)^{-3} = 0}$   $\ans{\displaystyle{x=-3}}$

    4. $\displaystyle{4(3x+2)^3 \cdot 3(x+5)^{-3} - 3(x+5)^{-4} \cdot (3x+2)^4 = 0}$   $\ans{\displaystyle{x=-\frac{2}{3},-18}}$

    5. $\displaystyle{5x^3(3x+1)^{2/3} + 3x^2(3x+1)^{5/3} = 0}$   $\ans{\displaystyle{x=0,-\frac{3}{14},-\frac{1}{3}}}$