## 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. $\cfrac{4 \pm \sqrt{31}}{5}$

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

4. For each function, use completing the square to find a formula for the function with only a single occurence of the variable. Then draw the graph of the function, indicating the coordinates of any interesting points (as always).

1. $f(x)=x^2+10x+20$

2. $g(x)=x^2-6x-5$

3. $h(x)=3x^2-7x-10$

4. $q(x)=-2x^2+4x-1$

5. $r(x)=x^2+x+1$

Click for full solutions to parts (a), (b), and (c)

The graphs for parts (d) and (e) can be found with Desmos.

Note, in part (d) the $x$-intercepts are $\left(\frac{2 \pm \sqrt{2}}{2},0\right)$, the $y$-intercept is at $((0,-1)$, and $(0,0)$ on the graph of $y=x^2$ moved under the transformations involved to $(1,1)$. This special point for quadratic functions is called the vertex of the graph.

In part (e), there are no $x$-intercepts, the $y$-intercept is at $(0,1)$, and the vertex is at $(-\frac{1}{2},\frac{3}{4})$.