QUANTITATIVE REASONING · GMAT-Q-AF

Algebraic Fractions

GMAT lesson development preview · DRAFT · Original learning and assessment material

Concepts and problem-solving strategy

Core mathematical relationship

\frac{x ^{2} - a ^{2}}{x - a} = x + a (x \neq = a)

Translate the condition

Factor the numerator as a difference of squares. Cancel only the common factor, and keep the original denominator restriction in view.

Test the result

Check the simplified result in the original expression. The excluded value still makes the original denominator zero.

Think Like a GMAT Test-Taker

Factor before cancelling, then record the values excluded from the original denominator.

Faster Methods

Translate the target into an equation before calculating.

Check the result against the original conditions and units.

Worked example 1

For x=3 and x not equal to 2, what is (x²−4)/(x−2)?

\frac{3 ^{2} - 4}{3 - 2} = \frac{5}{1} = 5

Factor the numerator as a difference of squares, cancel the common factor only because x is not 2, and then substitute x = 3 to get 5.

Worked example 2

Simplify the fraction with numerator x squared minus x minus 12 and denominator x squared minus 16, stating every excluded value.

\frac{x ^{2} - x - 12}{x ^{2} - 16} = \frac{( x - 4 ) ( x + 3 )}{( x - 4 ) ( x + 4 )} = \frac{x + 3}{x + 4}, x \neq = 4, - 4

First factor both polynomials. Cancel a common factor, not separate terms. The original denominator excludes both four and negative four; cancellation does not restore either input.

Worked example 3

A positive number x satisfies this equation: its reciprocal plus the reciprocal of the number one greater than x equals five sixths. Find x.

\frac{1}{x} + \frac{1}{x + 1} = \frac{5}{6} ⟹ 5 x^{2} - 7 x - 6 = 0 ⟹ (5 x + 3) (x - 2) = 0 ⟹ x = 2

Record that the denominators are nonzero, then multiply through by their product. Factor the resulting quadratic and reject the negative root because the problem requires a positive number. Substitute two into the original fractions to check.

Common Traps

Keep domain restrictions after simplifying.
Do not confuse the requested output with an intermediate quantity.