Digital SAT Math · Advanced Math

Rational equations and expressions

A rational equation is an ordinary equation wearing a trap. Somewhere in it a denominator is zero for one or two values of $x$, those values are never allowed, and the algebra you are about to do — multiply both sides by a common denominator — can spit those forbidden numbers back out as if they were solutions. The test knows this. The excluded value is almost always a printed choice, and “the equation has no solution” is waiting for the day every candidate fails the domain check. The whole skill is four steps with one rule that is not optional: write the excluded values before you clear anything, then throw them out again at the end.

On the test

Recognition cues: a variable under a fraction bar; two or more fractions set equal or combined; the phrases solution set, no solution, for all x such that; and any item that factors a quadratic denominator into linear pieces before you touch the numerators.

What this page owns

The College Board files rational equations under “nonlinear equations in one variable,” so most free pages bury them inside a five-family nonlinear lesson. We split them out. Here the defining move is clear a denominator and then discard forbidden roots. When the same factoring is used to rewrite an expression and nothing is ever solved, that is equivalent expressions. When the equation is a plain quadratic, a radical, or a product already equal to zero, that is nonlinear equations — which keeps one teaser rational item so the domain check is not a surprise the first time it appears. When the answer is a feature of a parabola, that is quadratic functions. The polynomial work is shared; the domain-first discipline is this page’s job.

Pattern recognition

Five shapes cover nearly every item:

1. Single fraction equals a constant — $\dfrac{ax+b}{cx+d} = k$, or $\dfrac{a}{x-r} = k$. One excluded value; one linear equation after clearing.
2. Two fractions set equal — cross-multiply (which is LCD multiply with two terms), watch both excluded values.
3. Sum or difference of fractions equals a third — LCD is usually the product of the linear denominators, or the factored form of a quadratic denominator already printed.
4. Polynomial over linear equals a linear or a constant — clearing produces a quadratic; one root is often the excluded value.
5. No solution — every candidate from the cleared equation is excluded, or the cleared equation is inconsistent after the domain cut.

Method

1. Note every excluded value. Any $x$ that zeros a denominator in the original equation is out. Factor quadratic denominators if needed so you can see the linear pieces.
2. Multiply through by the common denominator. Multiply every term — constants and lone variables included, not only the fractions. Distribute carefully through any parentheses that appear.
3. Solve the resulting polynomial equation. Linear ones finish in one line; quadratics factor or use the formula. You now have a list of candidates, not solutions.
4. Discard any candidate that was excluded. Substitute the survivors into the original equation (not the cleared one) to confirm. If the list is empty, the equation has no solution.

Practice

Answer before opening the explanation. Two items are student-produced response. At least three items print an extraneous root as a choice on purpose; “the equation has no solution” is a real answer on this skill, not a throwaway. Every wrong choice below is one named misconception — excluded value kept, constant term not multiplied by the LCD, minus distributed to only half a numerator, both candidates kept after a domain cut. When you miss one, log the misconception, not the item number.

Common mistakes

1. Picking the excluded value as the answer — it solves the cleared equation and makes the original undefined. Write the excluded list before clearing so the match is obvious at the end.
2. Keeping every root of the cleared equation — solution-set items put $\{a, b\}$ next to $\{a\}$ where $b$ is forbidden. Domain is a filter, not optional commentary.
3. Multiplying only the fraction terms by the LCD — a constant or a lone $x$ sitting next to a fraction still has to be multiplied. The partial clear produces a clean wrong linear root.
4. Sign error on a subtracted fraction — $-\dfrac{3}{x+2}$ times $(x-2)(x+2)$ is $-3(x-2) = -3x+6$. Writing $-3x-6$ flips the constant and lands on a tidy distractor.
5. Checking in the cleared equation — every candidate satisfies it by construction. Substitute into the equation as printed.
6. Calling a removable hole a solution — after cancelling a common factor $(x-r)$, the simplified equation may hold at $x = r$, but the original expression never had a value there.
7. Forgetting one factor of a quadratic denominator — $x^2 - 4 = (x-2)(x+2)$ excludes both $\pm 2$. Clearing by $x-2$ alone leaves a still-rational equation.
8. Reporting “no solution” too early — a quadratic can have two candidates of which only one is excluded. Discard the forbidden one; keep the survivor.
9. Reporting a solution when every candidate is excluded — the reverse of (8). Empty after the filter means no solution, not the excluded number.
10. Cross-multiplying a sum — $\dfrac{a}{b} + \dfrac{c}{d} = e$ is not a proportion. Build the LCD and multiply through; cross-multiply only when one fraction equals one fraction.
11. Answering a single root when the item asked for the solution set — and the reverse. Read the last words of the stem before choosing a letter.
12. Entering both roots on a grid-in that asked for the positive (or the greatest) one — the other root is often the excluded value or a negative the question already ruled out.

FAQ

Why can clearing a denominator create a fake solution? Because multiplying both sides by an expression is only reversible where that expression is not zero. At a zero of a denominator the original equation is undefined, so that input was never in the conversation — but the cleared polynomial does not know that and may list it as a root.

Is “no solution” the same as $x = 0$? No. Zero is a number that can solve an equation. “No solution” means the solution set is empty. On multiple choice it is a written option; a grid-in will not ask a question whose answer does not exist.

Do I always get a quadratic after clearing? No. Single-fraction and two-fraction proportions usually stay linear. Quadratics appear when a numerator is already degree 2, or when a constant multiplies a linear factor and you bring everything to one side. Degree is a signal, not a requirement.

Can I cancel before I solve? Yes, if you cancel a common factor of numerator and denominator and keep the restriction from that factor. Cancelling turns $\dfrac{(x-3)(x+2)}{x-3} = 5$ into $x+2 = 5$ with $x \neq 3$, which is often faster than expanding. Cancelling terms (pieces of a sum) is still illegal.

When is Desmos faster than hand clearing? When there are three different linear denominators or ugly coefficients. For a clean two-denominator item, writing the excluded list and multiplying is usually under thirty seconds and gives an exact fraction for free.

How is this different from radical extraneous roots? Same idea, different one-way step. Squaring destroys sign information; multiplying by a denominator invents roots at the zeros of that denominator. In both cases the cure is the same: test candidates in the original equation, and know why a particular candidate is likely to fail before you test it.

Grid-in format? Enter the exact value. Fractions are safer than rounded decimals (3/2 rather than 1.5). If the question asks for the positive solution and both a positive and a negative candidate appear, enter only the positive — and still confirm it is not excluded.

Editorial team · last reviewed 2026-08-12