How to Solve Power Equations

Power equations or quadratic equations without a first degree term are super fun to solve. You need to have control of roots to be able to find the answer without any aids. You also need to have control of the change side, change sign rule and how to get rid of a number in front of $x$.

Rule

PowerEquations

$\begin{array}{llll}\hfill a{x}^{2}+b& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =±\sqrt{-\frac{b}{a}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 1

$\begin{array}{llll}\hfill {x}^{2}& =49\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \sqrt{{x}^{2}}& =\sqrt{49}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =±7\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 2

$\begin{array}{llll}\hfill 2{x}^{2}& =98\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{2{x}^{2}}{2}& =\frac{98}{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {x}^{2}& =49\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \sqrt{{x}^{2}}& =\sqrt{49}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =±7\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

The root is such that $\sqrt{{a}^{2}}=a$. Any number raised to the power of 2 becomes a positive number, since $-×-=+$. This means that ${a}^{2}>0$, even if $a<0$.