Algebra 2 / Unit 4: Radical Functions and Rational Exponents
4.4 Solving Radical Equations
Notes
- Isolate the radical, then raise both sides to the index power ($\displaystyle \sqrt{\ }$: square; $\displaystyle \sqrt[3]{\ }$: cube).
- Solve the new equation.
- Always check in the original. Squaring can create extraneous solutions that don't work.
- Example: $\displaystyle \sqrt{x + 6} = x$ gives $x^2 - x - 6 = 0$, so $x = 3$ or $x = -2$. Check: $\displaystyle \sqrt{4} \ne -2$, so only $x = 3$.
- For $\displaystyle x^{\frac{m}{n}} = c$, raise both sides to $\displaystyle \frac{n}{m}$.
Practice
Solve $\displaystyle \sqrt{x + 3} = 5$.
Square: $x + 3 = 25$.
Solve $\displaystyle \sqrt{2x + 7} = x + 2$.
Squaring gives $x^2 + 2x - 3 = 0$, so $x = 1$ or $-3$. But $x = -3$ gives $1 = -1$, so it is extraneous.
Solve $\displaystyle x^{\frac{3}{2}} = 8$.
Raise to $\displaystyle \frac{2}{3}$: $\displaystyle x = 8^{\frac{2}{3}} = 4$.
Ready for more? Take the Unit 4 test.