Solving any equation by graphing

Any equation in one variable can be solved by graphing each side as its own curve and finding where they meet.

Example: to solve $\sqrt{x + 7} = x - 5$, graph $y = \sqrt{x+7}$ and $y = x - 5$. They cross only at $x = 9$.

• This automatically avoids extraneous solutions: squaring both sides by hand gives $x = 2$ too, but the graphs show it isn't real.
• Another way: move everything to one side and graph $y = (\text{left}) - (\text{right})$. The solutions are the $x$-intercepts.

On the test: use this for radical, absolute-value, and messy rational equations, where algebra is slow or error-prone.