56 Radical Equations

Radical Equations is the study of solving equations in which the variable appears under a radical, using the technique of raising both sides of the equation to a matching power to eliminate the radical, followed by a mandatory check of every resulting candidate solution against the original equation to discard any extraneous results introduced by that step.

The Scope of Radical Equations

A radical equation is an equation in which at least one side contains the variable under a radical, such as √(x + 3) = 5 or √x + 2 = x. Solving a radical equation requires eliminating the radical by raising both sides of the equation to a power matching the radical's index, converting the equation into an ordinary polynomial equation, which is then solved using established polynomial equation techniques.

Preparing a Radical Equation

Before eliminating a radical, the radical expression should be isolated on one side of the equation by itself, using ordinary addition, subtraction, multiplication, and division steps, exactly as isolating any other term in an equation. If more than one radical term is present, isolating one radical at a time, working through the equation in stages, is typically required.

Solving Single Even-Root Equations

Once a square root (or other even-indexed root) is isolated, both sides of the equation are raised to the power matching the root's index, eliminating the radical entirely.

The resulting equation is solved using ordinary linear or polynomial techniques, and the resulting candidate solution must be checked against the original radical equation before being accepted, since squaring both sides of an equation is not guaranteed to preserve the original equation's solution set.

Eliminating Repeated or Nested Radicals

Some radical equations require the raise-both-sides step to be applied more than once, when a second radical remains after the first is eliminated, or when the equation, after the first squaring, still contains a radical that must itself be isolated and eliminated in a second round of the same procedure.

Because squaring a binomial containing a radical does not remove that radical, the remaining radical term (2√x above) must be isolated and the entire equation squared a second time before a fully radical-free polynomial equation is reached.

Solving Equations with Odd Roots or Rational Powers

When an equation contains an odd-indexed root, such as a cube root, or a variable raised to a rational power, the same isolate-and-raise procedure applies, raising both sides to the power that matches the root's index or the reciprocal of the rational exponent. Because odd roots are defined for all real numbers, this case does not risk introducing the same type of extraneous solution associated with even roots, though checking the resulting candidate solution in the original equation remains good practice regardless.

Verifying Radical Equation Solutions

Every candidate solution obtained from solving a radical equation, particularly one involving an even-indexed root, must be substituted back into the original, unaltered radical equation — not the squared version — to confirm it produces a true statement. This verification step is not optional: raising both sides of an equation to an even power can introduce extraneous solutions, values that satisfy the squared equation but fail the original, because squaring can make a false statement (such as a negative value equaling a positive one before squaring) appear true afterward.

Diagnosing Errors in Radical Equations

Common errors in this area include squaring both sides of an equation before the radical has been fully isolated, forgetting that squaring a binomial containing a radical still leaves a radical term requiring a second isolation-and-squaring cycle, accepting a candidate solution without checking it against the original radical equation, and misapplying the extraneous-solution check to odd-root equations where it is generally unnecessary, potentially discarding a valid solution out of unwarranted caution. Because the verification step exists specifically to catch the consequences of a non-equivalence-preserving operation, it should be treated as the mandatory final stage of every radical equation solved using an even-indexed root.