SAT Math — radical and rational equations: extraneous solutions

The Advanced Math domain of the digital SAT names radical and rational equations among the nonlinear equations in one variable that students must solve fluently. These equations look short, but the usual solving moves can mislead: squaring both sides can create a value that does not satisfy the original equation, and clearing denominators can produce a value that makes a denominator zero. This material trains the habit that prevents both errors — solve, then check every candidate in the original equation.

The radical questions start with isolating a square root before squaring, then move to an equation where one of two candidates must be rejected because a principal square root is never negative, an equation with two square roots that needs to be rearranged before squaring, and a cube-root equation, where cubing is reversible and a negative result is allowed. A context question uses a speed model with a square root to find a distance.

The rational questions clear denominators with a common multiple and ask what happens next: an equation whose only candidate is excluded and therefore has no solution, an equation where one of two candidates makes a denominator zero, and an equation that leads to a quadratic with one positive solution. Two questions put rational equations in context — combined work rates of two pumps and the average cost per item of a production run — and one asks for the value of a constant that leaves an equation with no solution at all.

The material offers a quiz of twelve four-option questions, each with an explanation that shows the method and names the slip behind each wrong option, and a set of flashcards that collects the rules to remember. The explanations show the checking step in full every time, because that step is where most points are lost on these questions.

The content is based on the College Board's published skill descriptions for the Advanced Math domain of the digital SAT Math section. It is independent practice and is not produced or endorsed by the College Board.

• Isolate a radical, square or cube both sides, and solve the resulting equation
• Check every candidate in the original equation and reject extraneous solutions
• Solve rational equations by clearing denominators and exclude values that make a denominator zero
• Recognize when a radical or rational equation has no solution
• Set up and solve work-rate and average-cost equations in context

Practice material written by Zestly, based on the College Board's published description of the digital SAT Math section (Advanced Math domain: nonlinear equations in one variable — radical and rational equations), as given on satsuite.collegeboard.org and in the Assessment Framework for the Digital SAT Suite.