Exercises - Quadratic Equations, Pen-Sho, and Depressed Terms

  1. Solve the following equations:

    1. $\displaystyle{3x^2 - 4x - 2 = 0}$

    2. $\displaystyle{z^2 + 8z - 3 = 0}$

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

    4. $\displaystyle{5x^2 = 13 - 2x}$

    5. $\displaystyle{x(5x-2)=4}$

    6. $\displaystyle{2x(x+2) = 3}$

    1. $\displaystyle{x = \frac{2 \pm \sqrt{10}}{3}}$

    2. $\displaystyle{z = -4 \pm \sqrt{19}}$

    3. $\displaystyle{\frac{-5 \pm \sqrt{33}}{4}}$

    4. $\displaystyle{\frac{-1 \pm \sqrt{66}}{5}}$

    5. $\displaystyle{x = \frac{1 \pm \sqrt{21}}{5}}$

    6. $\displaystyle{\frac{x = -2 \pm \sqrt{10}}{2}}$

  2. Solve the following equations using all 4 methods (i.e., completing the square, Po-Shen Loh's method, the method of depressed terms, and the quadratic equation. If any solutions look different from one another, show they are actually the same solution.

    1. $3x^2 + 6 = 10x$

    2. $3t^2 + 8t + 3 = 0$

    3. $5t^2 - 8t = 3$

    4. $5m^2 + 3m = 2$

    5. $x^2 - 6x + 3 = 0$

    1. $\cfrac{5 \pm \sqrt{7}}{3}$

    2. $\cfrac{-4 \pm \sqrt{7}}{3}$

    3. $\cfrac{4 \pm \sqrt{32}}{5}$

    4. $-1,\cfrac{2}{5}$

    5. $3 \pm \sqrt{6}$

  3. Solve the following equations.

    1. $x - 3\sqrt{x} -4 = 0$

    2. $x^{2/3} + x^{1/3} - 6 = 0$

    3. $(2x-3)^2 - 5(2x-3) + 6 = 0$

    4. $(2t^2 + t)^2 - 4(2t^2 + t) + 3 = 0$

    5. $\displaystyle{x^4 - 9 = -2x^2}$

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

    7. $\displaystyle{\frac{x}{x+2} + \frac{2}{x-3} = \frac{10}{x^2 - x - 6}}$

    8. $2^x + 2^{-x} = 3$

    1. $16$

    2. $-27,8$

    3. $\frac{5}{2},3$

    4. $-\frac{3}{2},-1,\frac{1}{2},1$

    5. $\displaystyle{x = \pm \sqrt{-1 + \sqrt{10^{\phantom{1}}}}}$

    6. $\displaystyle{x = 3}$

    7. no solutions

    8. $\displaystyle{x = \log_2 \left(\cfrac{3 \pm \sqrt{5}}{2}\right)}$