QUANTITATIVE REASONING · GMAT-Q-QUAD

Quadratic Expressions and Equations

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

Concepts and problem-solving strategy

Model the relationship

x^{2} - S x + P = (x - r) (x - s), r + s = S, r s = P

Identify constraints and signs

x (x - k) = 0 \Rightarrow x = 0 or x = k

Verify with substitution

r^{2} + s^{2} = (r + s)^{2} - 2 r s

Think Like a GMAT Test-Taker

Move all terms to one side, factor if practical, and retain every root until the prompt's restrictions select the valid answer.

Faster Methods

Use the sum and product of roots when individual roots are unnecessary.

After dividing by a variable, check whether zero was a possible solution.

Underline the requested expression. Finding x is only an intermediate step if the question asks for x+3 or a product.

Worked example 1

If x²−5x−14=0 and x>0, find x.

(x - 7) (x + 2) = 0 \Rightarrow x = 7, - 2; x > 0 \Rightarrow x = 7

Both roots solve the equation; the stated domain selects seven.

Worked example 2

The roots of x²−9x+14=0 are r and s. Find r²+s².

r + s = 9, r s = 14; r^{2} + s^{2} = 81 - 28 = 53

Use the coefficients directly instead of computing and squaring each root.

Worked example 3

Solve x²=6x.

x (x - 6) = 0 \Rightarrow x = 0 or 6

Dividing immediately by x would lose the valid zero solution.

Worked example 4

If x is positive and x(x+3)=40, find x+3.

x^{2} + 3 x - 40 (x + 8) (x - 5) x > 0 x + 3 = 0 = 0 \Rightarrow x = 5 = 8

Move all terms to one side and find factors whose product is −40 and sum is 3. Both roots solve the quadratic, but −8 violates the domain. The question asks for x+3, so selecting five would stop one step too early.

Common Traps

Factoring gives alternatives: either factor can be zero.
A positive-root condition must be applied after solving, not assumed.