QUANTITATIVE REASONING · GMAT-Q-EXP

Expressions and Simplification

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

Concepts and problem-solving strategy

Model the relationship

a (b + c) = ab + a c

Identify constraints and signs

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

Verify with substitution

(a - b)^{2} = (a + b)^{2} - 4 ab

Think Like a GMAT Test-Taker

Look for the expression being requested before solving for individual variables. Combine like terms and use identities to expose quantities already given.

Faster Methods

Translate a square of a sum into the sum of squares plus twice the product.

When a linear combination is given, rewrite the target as a multiple of that combination.

Check whether the target is symmetric in the variables. A sum, product or squared difference may be determined even when the individual variables are not.

Worked example 1

If a+b=9 and ab=14, find a²+b².

a^{2} + b^{2} = (a + b)^{2} - 2 ab = 81 - 28 = 53

The requested symmetric expression can be found without solving for either variable.

Worked example 2

If 2x−3y=7, find 6x−9y+4.

6 x - 9 y + 4 = 3 (2 x - 3 y) + 4 = 25

Recognize the target as three times the given expression, plus four.

Worked example 3

Simplify 5(2x−3)−2(x−4).

10 x - 15 - 2 x + 8 = 8 x - 7

Distribute the negative coefficient to both terms before collecting like terms.

Worked example 4

If x+y=12 and x²+y²=80, find (x−y)².

144 (x - y)^{2} = 80 + 2 x y = 80 - 2 (32) x y = 32 = 16

First use the square of the sum to recover the product. Then use the square of the difference. The variables could appear in either order, but the requested squared difference is unchanged. Solving separately for x and y is unnecessary.

Common Traps

The square of a sum contains a cross term.
A negative sign before parentheses changes every term inside.