QUANTITATIVE REASONING · GMAT-Q-FACT

Factoring and Algebraic Identities

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

Concepts and problem-solving strategy

Model the relationship

a^{2} - b^{2} = (a - b) (a + b)

Identify constraints and signs

a x + a y = a (x + y)

Verify with substitution

\frac{x ^{2} - c ^{2}}{x - c} = x + c, x \neq = c

Think Like a GMAT Test-Taker

Identify a common factor or a recognizable identity before expanding. Keep track of values excluded by the original expression.

Faster Methods

Compute differences of squares as a product of a sum and a difference.

Factor a numerator before canceling common factors with a denominator.

Verify the simplified result using a permitted input in both the original and simplified expressions; never substitute an excluded value.

Worked example 1

Calculate 103²−97² without squaring both numbers.

(103 - 97) (103 + 97) = 6 (200) = 1200

The difference-of-squares identity makes the arithmetic short and exact.

Worked example 2

For x≠5, simplify (x²−25)/(x−5).

\frac{( x - 5 ) ( x + 5 )}{x - 5} = x + 5, x \neq = 5

Cancel the common factor; the original domain restriction remains.

Worked example 3

If a+b=13 and a−b=3, find a²−b².

a^{2} - b^{2} = (a - b) (a + b) = 3 (13) = 39

The requested product is already determined; finding a and b separately adds work.

Worked example 4

For a real number x≠3, if (x²−9)/(x−3)=2x−1, find x.

\frac{( x - 3 ) ( x + 3 )}{x - 3} x + 3 x = 2 x - 1, x \neq = 3 = 2 x - 1 = 4

Factor before canceling. The cancellation is valid only because x≠3. Solve the resulting linear equation and check the original expression: (16−9)/(4−3)=7, which equals 2×4−1. The restriction must survive simplification.

Common Traps

Cancel factors, not separate terms in a sum.
Simplifying an expression does not restore values excluded by its original denominator.