GRE Quantitative Comparison: Algebra Questions

Algebra Quantitative Comparison questions are among the most frequently tested QC types on the GRE. They ask you to compare two quantities involving algebraic expressions, equations, or inequalities and select one of four fixed answer choices. The twist is that answer D — "the relationship cannot be determined" — shows up more often in algebra QC than in any other domain, because variables with loose constraints can produce different comparison outcomes for different valid values. Below you will learn eight stem patterns the GRE uses, five strategies for resolving comparisons efficiently, and then practice with six interactive questions drawn from a curated question bank.

What Are QC Algebra Questions?

In a Quantitative Comparison question, you are shown two quantities — Quantity A and Quantity B — and sometimes additional information (constraints, equations, or conditions on variables) above them. Your job is to determine the relationship between the two quantities and select one of four fixed answer choices:

The four QC answer choices — these never change.

Choice	Meaning
A	Quantity A is greater.
B	Quantity B is greater.
C	The two quantities are equal.
D	The relationship cannot be determined from the information given.

When the underlying math involves algebraic concepts — equations, inequalities, expressions with variables, factoring, coordinate geometry, slopes, or word problems translated into algebraic form — you have an algebra QC question. There is no partial credit, and you select exactly one answer.

Key fact: Algebra QC questions are the most likely of all domains to have answer D. This is because they frequently involve variables with constraints that allow multiple values, and different values can produce different comparison outcomes. Do not default to D blindly, though — many algebra QC questions resolve definitively to A, B, or C after simplification.

Eight Stem Patterns You Will See

Nearly every algebra QC question on the GRE falls into one of eight recognizable patterns. Identifying the pattern quickly tells you which strategy to deploy.

Single Equation, Two Unknowns

A single linear equation with two variables and no additional constraints. Without more information, the relationship usually cannot be determined. Example:

x + y = -1

, compare

x

vs.

y

Inequality Constraint

One or more inequality constraints on variables. The answer depends on whether the comparison changes within the allowed range. Test boundary values strategically.

Product/Quotient Sign Constraint

Constraints on products or quotients of variables. Determine the sign of each variable individually. For example, if

x^2 \cdot y > 0

, then

y

must be positive.

Order Constraint

Variables are ordered (

r < s < t

) or defined as consecutive integers. Express all quantities in terms of one variable and simplify.

Quadratic/Polynomial Expression

Compare algebraic expressions that can be resolved through factoring or recognizing identities. Look for perfect squares and sum/difference of cubes.

Function/Slope Comparison

Compare slopes, intercepts, or function values using coordinate geometry concepts. Often requires finding the equation of a line through given points.

Algebraic Identity

Both quantities appear different but simplify to the same expression after distributing, factoring, or combining like terms. The answer is C.

Word Problem with Algebraic Translation

A real-world scenario requires translating the setup into algebra before comparing. Define variables clearly and set up the equation from the given information.

How to Solve QC Algebra Step by Step

These five strategies cover every algebra QC pattern. Master them in order and you will handle the vast majority of questions efficiently.

When variables appear with constraints, test specific values systematically. Start with simple values: 0, 1, -1, 1/2. Then try extremes: 10 or 100. If two different valid values produce different comparison outcomes, the answer is D. If all tested values give the same result, try to confirm algebraically.

Value to Test	Why It Matters
$x = 0$	Zeroes out terms; reveals whether the comparison depends on x at all
$x = 1$	Simplest positive value; good starting point
$x = -1$	Tests negative behavior; critical for sign-dependent comparisons
$x = \frac{1}{2}$	Tests fractions between 0 and 1 where $x^2 < x$
$x = -\frac{1}{2}$	Tests negative fractions
$x = 10$ or $100$	Tests large-number behavior

Instead of computing each quantity separately, compare them directly: compute Quantity A minus Quantity B (or vice versa). If the difference is always positive given the constraints, A is greater. If the difference is always zero, they are equal. If the difference can be positive or negative, the answer is D. Factor the difference expression whenever possible.

When the constraint involves a product or quotient of variables, determine the sign of each variable individually. Since $x^2$ is always non-negative, the condition $x^2 \cdot y > 0$ tells you $y$ is positive (and $x$ is nonzero). Similarly, $x \cdot y^2 < 0$ tells you $x$ is negative. This technique converts product constraints into definite sign information.

Look for common simplifications. The identity $(a+b)^2 = a^2 + 2ab + b^2$ lets you express $a^2 + b^2$ as $(a+b)^2 - 2ab$ . The sum-of-cubes formula $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ simplifies many fraction comparisons. If both sides reduce to the same expression, the answer is C.

For polynomial equations like $r^3 - r^2 - r + 1 = 0$ , factor to find all solutions. If different solutions produce different comparison outcomes, the answer is D. Be especially careful with quadratics — check the discriminant to determine whether there are one or two distinct roots.

Pro tip: When you compute Quantity A minus Quantity B and get an expression like

(x-1)^2 + 1

, recognize that this is always positive regardless of x. A perfect square plus a positive constant is always positive, giving you answer A immediately.

Worked Example: Symmetric Points

Work through each step below. You must answer each mini-challenge correctly to unlock the next step. If you get stuck, a second wrong attempt will reveal the answer so you can keep going.

Interactive Walkthrough0/4 steps

Comparing a Quadratic at Symmetric Points

f(x) = x^2 - 8x + 20

for all real numbers

x

Quantity A

Quantity B

f(2)

f(6)

Quantity A is greater

Quantity B is greater

The two quantities are equal

Cannot be determined

Step 1: Compute f(2)

Substitute

x = 2

into

f(x) = x^2 - 8x + 20

f(2) = 4 - 16 + 20

. What is

f(2)

Step 2: Compute f(6)

Step 3: Compare the quantities

Step 4: Understand why they're equal

Worked Example: Vieta's Formulas

This worked example demonstrates the subtraction strategy combined with completing the square — a powerful technique for QC questions involving polynomial expressions.

Interactive Walkthrough0/4 steps

Using Vieta's Formulas in a Comparison

The equation

2x^2 - 14x + c = 0

has two real roots

p

and

q

, where

c = 20

Quantity A

Quantity B

p + q

p \cdot q

Quantity A is greater

Quantity B is greater

The two quantities are equal

Cannot be determined

Step 1: Recall Vieta's formulas

For

ax^2 + bx + c = 0

, the sum of roots equals

-b/a

and the product of roots equals

c/a

. What is

p + q

for

2x^2 - 14x + 20 = 0

Step 2: Compute product of roots

Step 3: Compare the quantities

Step 4: Verify roots exist

Practice Questions

Test your skills with these six QC algebra questions. Each one uses a different pattern from the guide. After submitting your answer, walk through the step-by-step solution.

Practice Question 1 — Hard

a + b = 10

and

ab = 20

Quantity A

Quantity B

a^2 + b^2

2ab + 12

Select one answer:

Practice Question 2 — Hard

0 < x < y < z

and

x + y + z = 1

Quantity A

Quantity B

xz + y^2

xy + yz

Select one answer:

Practice Question 3 — Hard

3x + 5y = 30

, where

x

and

y

are positive numbers.

Quantity A

Quantity B

2x + y

Select one answer:

Practice Question 4 — Hard

t

is a real number such that

t^3 > t^2 > t

Quantity A

Quantity B

t^4 - t^3

t^2 - t

Select one answer:

Practice Question 5 — Hard

a^2 + b^2 = 13

and

a^4 + b^4 = 97

Quantity A

Quantity B

(ab)^2

Select one answer:

Practice Question 6 — Hard

0 < c < 1

Quantity A

Quantity B

\frac{c+1}{c^2+1}

Select one answer:

Three Common Traps

These traps catch test-takers repeatedly. Being aware of them before test day can save you from losing easy points.

Trap 1: The One-Value Fallacy

You test x = 1 and find that Quantity A is greater, so you pick A. But if you had also tested x = -1, you would have found Quantity B is greater — making the answer D. A single test value is never sufficient for a QC question involving variables. Always test at least two values, ideally from different regions of the constraint (positive, negative, fraction, large number).

Rule of thumb: If your first test value gives A greater and you are tempted to stop, force yourself to test one more value — especially a negative or a fraction. If the comparison flips, the answer is D.

Trap 2: Variables Can Be Fractions

Unless the problem explicitly states that variables are integers, they can be fractions, zero, or negative. The behavior of fractions between 0 and 1 is particularly deceptive: $x^2$ is less than $x$ (not greater), and $\frac{1}{x}$ is greater than $x$ . Many QC algebra questions are designed to exploit this — if you only test integers, you may miss a comparison reversal.

Trap 3: Canceling Variables Without Checking Signs

You cannot divide both sides of an inequality by a variable unless you know its sign. If x could be negative, dividing by x would flip the inequality. Instead, subtract one quantity from the other and factor. For example, rather than dividing both sides by x, compute A - B and determine whether the expression is positive, negative, or indeterminate.

Pattern Recognition Quick Reference

Use this table as a quick reference during practice. When you see a particular setup, jump to the corresponding strategy.

You See...	Strategy	Likely Answer
Single equation, two unknowns, no extra constraint	Test two values on opposite sides	D
Inequality constraint (e.g., $y > 4$ )	Subtract quantities and simplify	A, B, or D
Product/quotient constraint (e.g., $xy < 0$ )	Sign analysis on each variable	A or B
Ordered variables ( $r < s < t$ )	Express all in terms of one variable	A, B, or C
Polynomial expression, no constraint	Subtract and complete the square	A or B
Both quantities look different but complex	Simplify both — check for algebraic identity	C
Polynomial equation (e.g., cubic = 0)	Factor to find all roots, test each	D
Word problem with unknowns	Translate to algebra, then apply other strategies	Varies

Remember: The table above shows likely answers, not guaranteed answers. Always verify with the actual problem constraints. A question that looks like Pattern 1 may have an extra constraint that makes the answer A, B, or C.

Study Checklist

Track your preparation with this interactive checklist. Click each item as you complete it.

QC Algebra Mastery Checklist0/8 complete

Frequently Asked Questions

How common are algebra-based Quantitative Comparison questions on the GRE?

Algebra QC questions are among the most common QC types on the GRE. Roughly one-third to one-half of all Quantitative Comparison questions involve algebraic reasoning such as equations, inequalities, or expressions with variables. Expect two to four algebra-based QC questions per Quantitative section.

Why is answer D so common in QC algebra questions?

Answer D appears frequently because algebra QC questions often involve variables with loose constraints that allow multiple valid values. When different valid values produce different comparison outcomes, the answer must be D. A single equation with two unknowns, for example, almost always leads to D because infinitely many pairs of values satisfy the equation.

What is the most important strategy for QC algebra questions?

Plugging in values is the single most important strategy. Test at least two or three values that satisfy the given constraints, including edge cases like 0, 1, -1, and fractions. If two different valid values produce different comparison outcomes, the answer is D. If all tested values give the same result, try to confirm algebraically by computing the difference A - B and factoring.

Can I multiply or divide both quantities by a variable in a QC question?

You can only multiply or divide both quantities by a positive value without changing the comparison. If the variable could be negative, multiplying or dividing would flip the inequality. If you do not know the sign of a variable, do not multiply or divide by it. Instead, subtract one quantity from the other and analyze the sign of the difference.

Should I always try to simplify algebraically before plugging in values?

Ideally, start with algebraic simplification. Compute Quantity A minus Quantity B and try to factor or recognize identities. If the difference is always positive, always negative, or always zero, you have a definitive answer without plug-in. But if you cannot simplify cleanly, switch to strategic plug-in immediately. Many Hard QC questions are designed so that algebraic simplification reveals the answer quickly, saving you time compared to testing multiple values.