Question

# 25p² - 10 p = 2

$$\displaystyle{25}{p}²-{10}{p}={2}$$

2021-02-26
We are given: $$\displaystyle{25}{p}²-{10}{p}={2}$$
Solve by completing the square. First, we write the equation in the form $$\displaystyle{x}²-{10}{x}={c}$$. Divide both sides by 25:
$$\displaystyle{p}²-\frac{{2}}{{5}}{p}=\frac{{2}}{{25}}$$
Complete the square by adding $$\displaystyle{\left(\frac{{b}}{{2}}\right)}^{{2}}$$ on both sides. Here, $$\displaystyle{b}=-{\left(\frac{{2}}{{5}}\right)}$$ so we add $$\displaystyle{\left(\frac{{-\frac{{2}}{{5}}}}{{2}}\right)}^{{2}}=\frac{{1}}{{25}}$$ to both sides:
$$\displaystyle{p}²-\frac{{2}}{{5}}{p}+\frac{{1}}{{25}}=\frac{{2}}{{25}}+\frac{{1}}{{25}}$$
$$\displaystyle{\left({p}-{\left(\frac{{1}}{{5}}\right)}\right)}^{{2}}=\frac{{3}}{{25}}$$
Take the square root of both sides: $$\displaystyle{p}-{\left(\frac{{1}}{{5}}\right)}=\pm\frac{\sqrt{{3}}}{{5}}$$
Add 1/5 to both sides: $$\displaystyle{p}=\frac{{1}}{{5}}\pm\frac{\sqrt{{3}}}{{5}}$$
$$\displaystyle{p}=\frac{{{1}\pm\sqrt{{3}}}}{{5}}$$