Radical Expressions and Equations
What you must understand
A radical expression involves roots, most often square roots. Simplifying means removing perfect-square factors from the radicand: sqrt(72) = 6*sqrt(2). Radicals can be added only when they are like radicals with the same index and radicand. Multiplying uses the product property, and rationalizing a denominator removes the radical from the bottom, multiplying by the conjugate when the denominator is a binomial. Radical equations are solved by isolating the radical and raising both sides to the power of the index. That operation can create extraneous solutions, so every answer must be substituted back into the original equation. Remember that the square root symbol denotes the principal (nonnegative) root, so sqrt(x^2) = |x|, not x.
Traps to watch
- Writing sqrt(a + b) = sqrt(a) + sqrt(b), which is false.
- Forgetting to check answers after squaring, admitting extraneous roots.
- Squaring a binomial incorrectly by squaring each term separately.
- Ignoring the absolute value in sqrt(x^2).
- Rationalizing by multiplying by the conjugate of the numerator instead of the denominator.
Worked example
Solve sqrt(2x + 3) = x. Square both sides: 2x + 3 = x^2, so x^2 - 2x - 3 = 0, giving (x - 3)(x + 1) = 0 and x = 3 or x = -1. Check: x = 3 gives sqrt(9) = 3, valid. x = -1 gives sqrt(1) = 1, but the right side is -1, so -1 is extraneous. The solution is x = 3.
Rule of thumb
Isolate the radical, raise to the index power, then check every result in the original equation.