Radical equations are equations that contain a variable inside a root, such as a square root or cube root. They matter because roots appear in geometry, physics, statistics, and formulas involving distance, energy, and rates. Solving them requires careful algebra because removing a radical can change the equation.
The main goal is to isolate the radical, undo it with a power, and verify the result.
Key Facts
- A radical equation has a variable under a root, such as sqrt(x + 5) = 4.
- First isolate the radical before raising both sides to a power.
- For square roots, if sqrt(A) = B, then A = B^2, but B must be nonnegative.
- For cube roots, if cbrt(A) = B, then A = B^3.
- Squaring both sides can create extraneous solutions, so every proposed answer must be checked.
- Example pattern: sqrt(x + 3) = x - 1 gives x + 3 = (x - 1)^2, then solve and check.
Vocabulary
- Radical equation
- An equation in which the variable appears inside a radical expression such as a square root or cube root.
- Radical
- A symbol that represents taking a root, such as sqrt(x) for the square root of x.
- Index
- The small number on a radical that tells which root is being taken, such as 3 in a cube root.
- Extraneous solution
- A value that appears during algebraic solving but does not satisfy the original equation.
- Domain restriction
- A condition that limits which input values are allowed, such as requiring the inside of a square root to be nonnegative.
Common Mistakes to Avoid
- Squaring before isolating the radical is wrong because extra terms can make the algebra much harder or incorrect. Move constants and coefficients first so the radical stands alone.
- Forgetting to square the entire side is wrong because (x - 1)^2 is not x^2 - 1. Use parentheses around each full side before raising it to a power.
- Accepting every algebraic answer is wrong because squaring can create extraneous solutions. Substitute each candidate into the original radical equation.
- Ignoring square root restrictions is wrong because sqrt(A) is defined only when A is nonnegative in real-number algebra. Check the radicand and any isolated square-root expression before finalizing answers.
Practice Questions
- 1 Solve and check: sqrt(x + 7) = 5.
- 2 Solve and check: sqrt(2x - 3) = x - 3.
- 3 A student solves sqrt(x + 1) = x - 1 and finds x = 0 and x = 3 after squaring. Explain which value is extraneous and why.