GMAT algebra review covers the core topics that make up roughly 20% of the Quantitative Reasoning section — linear equations, quadratics, inequalities, and exponent rules. Combined with arithmetic, algebra accounts for over 51% of all quant questions, making it the single most important area to master for a high score.
GMAT linear equations appear 1 to 2 times per quant section, both in problem-solving format (typically as word problems) and in Data Sufficiency format (testing whether a system has a unique solution). These are foundational questions — getting them right quickly frees up time for harder problems.
Single-variable linear equations involve isolating the variable through inverse operations. The key principle is doing the same operation to both sides. While this seems basic, the GMAT tests this through word problems where translating the scenario into an equation is the real challenge — the algebra itself is straightforward once set up correctly.
When a GMAT problem involves two unknowns, you need two independent equations. The substitution method works best when one variable is easily isolatable: solve one equation for a variable, then plug it into the other. The elimination method works better when coefficients align — multiply one or both equations by constants to cancel a variable.
For Data Sufficiency, the critical insight is that you do not need to solve the system. Two independent linear equations with two unknowns always yield a unique solution. If both statements together give you two independent equations, the answer is determinable — stop there.
Problem: A company charges a flat fee plus a per-unit cost for printing. If 100 copies cost $250 and 300 copies cost $550, what is the flat fee?
The flat fee is $100. Classic GMAT word problem that translates directly into a system of linear equations solved by elimination.
The hardest part of GMAT linear equation problems is not the algebra — it is translating English into equations. Look for keywords: "total" signals addition, "per" signals multiplication, "more than" signals addition, and "times" signals multiplication. Define your variables clearly before writing any equation, and double-check that your equation matches the problem statement before solving.
GMAT quadratic equations follow the standard form ax² + bx + c = 0, and the vast majority of them are designed to be solved by factoring — not by the quadratic formula. A test-taker can break 700 or even 750 without using the quadratic formula even once, because factoring is faster and less error-prone.
To factor a quadratic like x² + bx + c = 0, find two numbers that multiply to c and add to b. For example, x² - 5x + 6 = 0 factors as (x - 2)(x - 3) = 0 because -2 and -3 multiply to 6 and add to -5. Set each factor to zero to find x = 2 or x = 3. Practice this until you can do it in under 20 seconds — speed on these problems comes entirely from pattern recognition.
Solve: x² - 5x + 6 = 0
The roots are x = 2 and x = 3. On the GMAT, this factoring approach takes under 20 seconds — far faster than the quadratic formula.
The discriminant D = b² - 4ac tells you about the nature of a quadratic's roots without solving. If D > 0, the equation has two distinct real roots. If D = 0, it has one repeated root. If D < 0, it has no real roots (complex numbers, which the GMAT does not test). This is especially useful in Data Sufficiency, where determining whether roots exist is often more important than finding them.
Inequality problems are where many GMAT test-takers lose points — not because the concepts are hard, but because the rules for manipulating GMAT inequalities differ subtly from equation rules in ways that are easy to forget under pressure.
You can add, subtract, multiply, or divide both sides of an inequality by a positive number without changing the direction of the sign. But when you multiply or divide by a negative number, you must flip the inequality sign. This is the single most common inequality error on the GMAT. For compound inequalities, you can combine them only when the signs face the same direction.
The absolute value |x| represents the distance from zero, which is always non-negative. For GMAT purposes, you need two key patterns. When |expression| < k (where k > 0), rewrite as a compound inequality: -k < expression < k. When |expression| > k, split into two separate inequalities: expression > k OR expression < -k.
Solve: |2x - 6| < 10
The solution is -2 < x < 8. Absolute value inequalities of the form |expression| < k always translate into a range.
Exponent rule confusion is one of the most common sources of GMAT algebra errors. The rules themselves are simple, but under time pressure students frequently confuse which operation applies. The solution is to have the rules so thoroughly memorized that applying them becomes automatic.
| Rule | Formula | Example |
|---|---|---|
| Product Rule | a^m × a^n = a^(m+n) | 2³ × 2⁴ = 2⁷ = 128 |
| Power Rule | (a^m)^n = a^(m×n) | (2³)⁴ = 2¹² = 4096 |
| Quotient Rule | a^m ÷ a^n = a^(m-n) | 2⁵ ÷ 2² = 2³ = 8 |
| Zero Exponent | a⁰ = 1 | 5⁰ = 1 |
| Negative Exponent | a^(-n) = 1/a^n | 2^(-3) = 1/8 |
| Fractional Exponent | a^(1/n) = ⁿ√a | 8^(1/3) = ³√8 = 2 |
Roots are simply fractional exponents: the square root of a equals a^(1/2), the cube root equals a^(1/3), and so on. This means all exponent rules apply to roots as well. For example, the square root of (a × b) equals the square root of a times the square root of b. But critically, the square root of (a + b) does NOT equal the square root of a plus the square root of b — this is one of the most common distributing errors on the GMAT.
Three algebraic patterns appear so frequently on the GMAT that recognizing them instantly saves substantial time. These are not obscure formulas — they are the backbone of efficient GMAT algebra solving.
| Formula Name | Formula | Example |
|---|---|---|
| Difference of Two Squares | a² - b² = (a - b)(a + b) | x² - 9 = (x - 3)(x + 3) |
| Square of a Sum | (a + b)² = a² + 2ab + b² | (x + 4)² = x² + 8x + 16 |
| Square of a Difference | (a - b)² = a² - 2ab + b² | (x - 5)² = x² - 10x + 25 |
| Discriminant | D = b² - 4ac | For x² + 3x + 2: D = 9 - 8 = 1 (two real roots) |
| Slope-Intercept Form | y = mx + b | y = 2x + 3 (slope 2, y-intercept 3) |
The GMAT tests coordinate geometry as an extension of algebra. Every line can be expressed as y = mx + b, where m is the slope and b is the y-intercept. The slope between two points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1). Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals of each other.
The incorrect distribution of operations over addition and subtraction is probably the single greatest source of algebra errors on the GMAT. Understanding these common mistakes and building awareness of them is as valuable as learning the formulas themselves.
| Mistake | What Students Do | Correct Approach |
|---|---|---|
| Distributing roots | √(a + b) = √a + √b | Cannot simplify √(a + b); factor inside the root if possible |
| Confusing exponent rules | a^m × a^n = a^(m×n) | Correct: a^m × a^n = a^(m+n) — add exponents for multiplication |
| Dropping the negative | If x = -5, then -x = -5 | -x means -(x), so -(-5) = 5 — two negatives make a positive |
| Canceling without factoring | Cancel terms across + or - | Factor first, then cancel common factors from numerator and denominator |
| Solving DS completely | Find x = 3 in Data Sufficiency | Only determine whether x can be uniquely determined, then stop |
In Data Sufficiency, the question is not "What is the answer?" but "Can the answer be determined?" Many students waste time solving equations completely when they only need to verify that a unique solution exists. For systems of linear equations, simply count: if you have as many independent equations as unknowns, a unique solution exists. For quadratics, check the discriminant rather than finding the roots.
Test your understanding of the algebra concepts covered above.
Enter the coefficients of a quadratic equation (ax² + bx + c = 0) to find the discriminant and determine the nature of its roots.