Integrated Math 3 / Unit 4: Radical Functions and Equations
4.4 Solving Radical Equations
Notes
To solve a radical equation, isolate the radical, then raise both sides to the matching power.
- Square both sides for a square root, cube both sides for a cube root.
- $\displaystyle \sqrt{x + 7} = 5 \Rightarrow x + 7 = 25 \Rightarrow x = 18$.
- $\displaystyle \sqrt[3]{2x + 3} = -1 \Rightarrow 2x + 3 = -1 \Rightarrow x = -2$.
- With a rational exponent, raise both sides to the reciprocal power: $x^{3/2} = 8 \Rightarrow x = 8^{2/3} = 4$.
- Always check your answer in the original equation.
Practice
Solve $\displaystyle \sqrt{x + 7} = 5$.
Square both sides: $x + 7 = 25$.
Solve $\displaystyle \sqrt[3]{2x + 3} = -1$.
Cube both sides: $2x + 3 = -1$, so $2x = -4$.
Solve $x^{3/2} = 8$.
Raise both sides to the $\displaystyle \frac{2}{3}$ power: $x = 8^{2/3} = 4$. Check: $4^{3/2} = 8$.
Ready for more? Take the Unit 4 test.