The recurring algebraic principles that make GMAT quant questions easier to reason through.
Algebraic expressions almost always have a common factor sitting inside them, and pulling it out before doing anything else exposes the structure the rest of the work depends on. The instinct should be automatic: scan first, factor, then decide on a method. Most of the time, the factored form makes the next step obvious; expanding or solving before factoring usually means working with terms that were going to cancel anyway.
The difference of two squares — a² − b² = (a + b)(a − b) — is the single most useful identity in algebra, and the GMAT hides it everywhere. a⁴ − b² is really (a²)² − b². 5¹⁶ − 3¹⁶ is the same trick. Even fractions like (x² − 1)/(x + 1) collapse the moment you spot the pattern. Train the eye to register two squares being subtracted, in any disguise.
The question dictates the work — and the GMAT routinely asks for a combination like x + y rather than for x and y separately. Reading carefully enough to spot the distinction often unlocks shortcuts: factoring may expose the requested combination directly, even when solving for each variable individually is impossible. Mismatching the work to the question is one of the most common ways to waste time on a problem.
Dividing both sides of an equation by a variable looks like progress, but it silently loses the solution where that variable equals zero. The safe alternative is to bring everything to one side and factor instead. Where division gives one answer, factoring usually gives two — and the missing solution is often the one the question wanted.
When solving simultaneous equations, the question almost always asks for one specific variable. Decide which variable you want to keep, and choose the method that removes the others most efficiently. Without that clarity, students often eliminate the wrong variable or solve the entire system when only one quantity was needed — costing time and creating opportunities for error.
Expanding (ax + b)(cx + d) requires multiplying every term in the first bracket by every term in the second, then collecting like terms. A common error is squaring a sum term-by-term — (3x + 2)² is not 9x² + 4 but 9x² + 12x + 4. Always expand fully.
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Two identities worth knowing instinctively: (x + y)² = x² + 2xy + y² and (x − y)² = x² − 2xy + y² — they differ only in the middle sign. A useful consequence: (x + y)² − (x − y)² simplifies to 4xy, a shortcut the GMAT uses surprisingly often.
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Algebraic fractions only simplify after the numerator and denominator are factored. Once factored, shared factors cancel cleanly. Watch for hidden patterns — (x² − 1)/(x + 1) cancels to (x − 1) only because the numerator is really (x − 1)(x + 1). Factor first; cancel second.
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For equations of the form a/b = c/d, multiplying both sides by both denominators clears the fractions instantly: ad = bc. This converts a proportion into a clean polynomial equation in a single step, no working required. It's the fastest way to handle any GMAT equation with one fraction equal to another.
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When an equation contains fractions with variables in the denominator — 3/x + 5x + 5 = 0 — multiplying every term by the denominator clears the fraction outright. The remaining equation contains only polynomial terms and is much easier to solve. Use this whenever a variable appears in a denominator.
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A linear system of n unknowns needs exactly n genuinely different equations to have a unique solution. Fewer equations leaves the system underdetermined; more than n is usually consistent but redundant. The rule is most useful for data sufficiency, where you can declare a system solvable without actually solving it.
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Three traps break the n-equations-for-n-unknowns rule. A scaled duplicate — 2x + 4y = 8 of x + 2y = 4 — doesn't count as a second equation. Integer restrictions, like "x and y are positive integers," can pin down a system with fewer equations than unknowns. And a quadratic, though counted as one equation, may yield zero, one, or two solutions — only one of which counts as sufficient in data sufficiency.
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Even powers split into ± when rooted: x² = 4 gives x = ±2; the same applies to x⁴, x⁶, and so on. Odd powers preserve sign: x³ = 27 gives x = 3 only. An even power equal to a negative number has no real solution.
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Elimination cancels a variable by adding or subtracting scaled equations — multiply equation one by 2, subtract equation two, and the unwanted variable disappears. Division is a faster form when equations share multiplicative structure: if xyz = 10 and xy = 5, dividing one by the other gives z = 2 instantly. Both methods work only on linear systems.
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Substitution rearranges one equation to express a single variable, then plugs that expression into the others to reduce the unknown count. Unlike elimination, it works on any system — and is required the moment a quadratic or product appears, since elimination only handles linear equations.
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