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GMAT (Graduate Management Admission Test) Quantitative Reasoning: Algebra Syllabus
The Quantitative Reasoning: Algebra syllabus for GMAT (Graduate Management Admission Test) in full: 5 chapters, 25 topics and 10 sub-topics, every one listed below, plus 50 Quantitative Reasoning: Algebra flashcards with answers.
Quantitative Reasoning: Algebra syllabus: all 5 chapters
Expand any chapter to see its topics and sub-topics.
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Algebraic Expressions & Manipulation
5 topics- Combining like terms and the distributive law
- Factoring techniques
- Common factor extraction
- Special products (difference of squares, perfect squares)
- Factoring quadratics
- Polynomial multiplication (FOIL)
- Simplifying rational expressions
- Substitution and evaluating expressions
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Linear Equations & Inequalities
5 topics- Solving single-variable linear equations
- Systems of linear equations
- Substitution method
- Elimination method
- When systems have no or infinite solutions
- Linear inequalities and flipping the sign
- Compound and absolute-value inequalities
- Translating word problems into equations
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Quadratics & Higher-Order Equations
5 topics- Solving quadratics by factoring
- The quadratic formula and discriminant
- Completing the square
- Roots, sum and product of roots
- Equations reducible to quadratic form
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Functions, Formulas & Sequences
5 topics- Function notation and evaluation
- Symbolism / made-up function problems
- Arithmetic sequences
- Common difference and nth term
- Sum of an arithmetic series
- Geometric sequences
- Rearranging and solving formulas
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Coordinate Geometry (Algebra-based)
5 topics- The x-y plane and plotting points
- Slope of a line
- Positive, negative, zero, and undefined slope
- Parallel and perpendicular slopes
- Equations of lines (slope-intercept and point-slope)
- Distance and midpoint formulas
- Intercepts and intersection of lines
GMAT (Graduate Management Admission Test) Quantitative Reasoning: Algebra flashcards — 50 questions with answers
What does it mean to combine like terms, and which terms qualify as "like"?
Combining like terms means adding/subtracting the coefficients of terms that have the exact same variable part (same variables raised to the same powers). E.g., 3x + 5x = 8x, and 2x^2 + 4x cannot be combined.
State the distributive law and give an example.
a(b + c) = ab + ac. The factor outside is multiplied by every term inside the parentheses. Example: 3(x + 4) = 3x + 12.
What is the FOIL method for multiplying two binomials (a + b)(c + d)?
FOIL = First, Outer, Inner, Last: (a+b)(c+d) = ac + ad + bc + bd. Multiply the first terms, outer terms, inner terms, and last terms, then combine like terms.
Expand (x + y)^2, (x - y)^2, and (x + y)(x - y).
(x+y)^2 = x^2 + 2xy + y^2; (x-y)^2 = x^2 - 2xy + y^2; (x+y)(x-y) = x^2 - y^2 (difference of squares).
Factor the difference of squares a^2 - b^2.
a^2 - b^2 = (a + b)(a - b).
How do you factor a quadratic of the form x^2 + bx + c?
Find two numbers that multiply to c and add to b. If they are p and q, then x^2 + bx + c = (x + p)(x + q).
What is the first step in factoring any polynomial?
Factor out the greatest common factor (GCF) shared by all terms. E.g., 6x^2 + 9x = 3x(2x + 3).
Factor x^3 + x^2 by grouping/GCF and state the technique called grouping.
x^3 + x^2 = x^2(x + 1). Factoring by grouping pairs terms with common factors: e.g., ax + ay + bx + by = a(x+y) + b(x+y) = (a+b)(x+y).
How do you simplify a rational expression?
Factor the numerator and denominator completely, then cancel any common factors. E.g., (x^2 - 9)/(x + 3) = ((x+3)(x-3))/(x+3) = x - 3 (x ≠ -3).
What restriction must you keep in mind when simplifying rational expressions?
The denominator can never equal zero, so any value making the original denominator zero is excluded from the domain even after canceling.
What does "substitution" / "evaluating an expression" mean?
Replacing each variable with its given numerical value and then computing the result following order of operations (PEMDAS). E.g., for x=2, 3x^2 - 1 = 3(4) - 1 = 11.
List the order of operations (PEMDAS).
Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
What are the steps to solve a single-variable linear equation?
Simplify each side (distribute, combine like terms), move all variable terms to one side and constants to the other using inverse operations, then divide by the coefficient to isolate the variable.
How many solutions can a linear equation in one variable have?
Exactly one solution normally; no solution if it reduces to a false statement (e.g., 0 = 5); infinitely many solutions if it reduces to a true identity (e.g., 0 = 0).
What are the two main algebraic methods for solving a system of two linear equations?
Substitution (solve one equation for a variable and plug into the other) and elimination/combination (add or subtract scaled equations to cancel a variable).
How can you tell whether a 2x2 linear system has one, none, or infinitely many solutions?
One solution if the lines intersect (different slopes); no solution if the lines are parallel (same slope, different intercepts); infinitely many if the equations are the same line (same slope and intercept).
What is the key rule when multiplying or dividing an inequality by a negative number?
You must flip (reverse) the direction of the inequality sign. E.g., -2x < 6 becomes x > -3.
When does the inequality sign NOT flip while solving an inequality?
When adding/subtracting any number to both sides, or multiplying/dividing both sides by a positive number. The sign only flips for multiplication/division by a negative.
How do you solve a compound inequality such as -3 < 2x + 1 ≤ 7?
Perform the same operation on all three parts: subtract 1 to get -4 < 2x ≤ 6, then divide by 2 to get -2 < x ≤ 3.
How do you solve an absolute-value equation |x| = a (a > 0)?
Split into two cases: x = a or x = -a. If a < 0 there is no solution; if a = 0 then x = 0.
Show the other 30 Quantitative Reasoning: Algebra flashcards
How do you solve the absolute-value inequalities |x| < a and |x| > a (a > 0)?
|x| < a means -a < x < a (an AND/between case). |x| > a means x < -a OR x > a (an OR/outside case).
In word problems, how do you translate "is," "of," "more than," and "percent" into algebra?
"is" → equals (=); "of" → multiply (×); "more than" → add (+); "less than" → subtract; "percent" → divide by 100. E.g., "20% of x" = 0.20x.
What is the general process for translating a word problem into an equation?
Identify the unknown and assign a variable, express other quantities in terms of that variable, translate the relationships into an equation, solve, and check the answer against the problem's conditions.
How do you solve a quadratic by factoring, and what principle do you use?
Set the equation equal to 0, factor it, then use the Zero Product Property: if AB = 0 then A = 0 or B = 0. Solve each factor for the roots.
State the quadratic formula for ax^2 + bx + c = 0.
x = (-b ± √(b^2 - 4ac)) / (2a), where a ≠ 0.
What is the discriminant, and what does its value tell you?
The discriminant is D = b^2 - 4ac. If D > 0 there are two distinct real roots; if D = 0 there is one repeated real root; if D < 0 there are no real roots (two complex roots).
What condition on the discriminant means a quadratic has exactly one (repeated) real solution?
b^2 - 4ac = 0, which corresponds to a perfect-square trinomial / the parabola being tangent to the x-axis.
Describe the steps to solve a quadratic by completing the square for x^2 + bx + c = 0.
Move c to the other side, add (b/2)^2 to both sides to form a perfect square, write the left side as (x + b/2)^2, then take the square root of both sides and solve.
For x^2 + bx, what term completes the square?
Add (b/2)^2. Then x^2 + bx + (b/2)^2 = (x + b/2)^2.
For ax^2 + bx + c = 0, what are the sum and product of the roots?
Sum of roots = -b/a; Product of roots = c/a.
If the roots of a quadratic are r and s, how can you write the quadratic (leading coefficient 1)?
x^2 - (r + s)x + rs = 0, i.e., x^2 - (sum)x + (product) = 0.
How do you solve an equation reducible to quadratic form, such as x^4 - 5x^2 + 4 = 0?
Substitute u = x^2 (so x^4 = u^2), giving u^2 - 5u + 4 = 0; solve for u, then back-substitute and solve x^2 = u for x. Here u = 1 or 4, so x = ±1, ±2.
What substitution helps solve an equation like x - 5√x + 6 = 0?
Let u = √x (so x = u^2), giving u^2 - 5u + 6 = 0; solve for u, then x = u^2. Check that u ≥ 0 since √x cannot be negative.
What does the function notation f(x) mean, and how do you evaluate f(3)?
f(x) is the output (value) of function f for input x. To find f(3), substitute 3 for every x in the rule. E.g., if f(x)=2x+1, then f(3)=7.
How do you evaluate a composite function f(g(x))?
Work from the inside out: first compute g(x), then use that result as the input to f. E.g., f(g(2)) means find g(2) first, then plug it into f.
In a "symbolism" or made-up-operator problem, how do you handle a defined operation like a ◊ b = a^2 - b?
Treat the symbol as a rule/recipe: substitute the given values into the definition exactly. E.g., 3 ◊ 5 = 3^2 - 5 = 9 - 5 = 4.
What is an arithmetic sequence, and what is its common difference?
A sequence where each term is obtained by adding a fixed number (the common difference, d) to the previous term. E.g., 2, 5, 8, 11 has d = 3.
State the formula for the nth term of an arithmetic sequence.
a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference.
State the formula for the sum of the first n terms of an arithmetic sequence.
S_n = n/2 · (a_1 + a_n) = n/2 · [2a_1 + (n - 1)d].
What is a geometric sequence, and what is its common ratio?
A sequence where each term is obtained by multiplying the previous term by a fixed number (the common ratio, r). E.g., 3, 6, 12, 24 has r = 2.
State the formula for the nth term of a geometric sequence.
a_n = a_1 · r^(n-1), where a_1 is the first term and r is the common ratio.
How do you rearrange a formula to solve for a specific variable (e.g., solve A = lw for w)?
Treat the target variable as the unknown and isolate it using inverse operations on both sides. For A = lw, divide both sides by l to get w = A/l.
How are points plotted on the x-y (Cartesian) coordinate plane?
A point is written as an ordered pair (x, y): move x units horizontally from the origin (right if positive, left if negative), then y units vertically (up if positive, down if negative).
Name the four quadrants of the coordinate plane by the signs of (x, y).
Quadrant I (+,+), Quadrant II (-,+), Quadrant III (-,-), Quadrant IV (+,-), numbered counterclockwise starting from the upper right.
State the slope formula between two points (x1, y1) and (x2, y2).
slope m = (y2 - y1) / (x2 - x1), i.e., rise over run (change in y divided by change in x).
What are the slopes of horizontal lines, vertical lines, parallel lines, and perpendicular lines?
Horizontal line: slope 0. Vertical line: slope undefined. Parallel lines: equal slopes. Perpendicular lines: slopes are negative reciprocals (m1 · m2 = -1).
State the slope-intercept form of a line and identify its parts.
y = mx + b, where m is the slope and b is the y-intercept (the y-value where the line crosses the y-axis).
State the point-slope form of a line.
y - y1 = m(x - x1), where m is the slope and (x1, y1) is a known point on the line.
State the distance formula and the midpoint formula for points (x1, y1) and (x2, y2).
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]. Midpoint = ((x1 + x2)/2, (y1 + y2)/2).
How do you find the x-intercept and y-intercept of a line, and how do you find where two lines intersect?
x-intercept: set y = 0 and solve for x; y-intercept: set x = 0 and solve for y. Intersection of two lines: solve the two equations simultaneously (substitution or elimination).
GMAT (Graduate Management Admission Test) Quantitative Reasoning: Algebra FAQ
What is in the GMAT (Graduate Management Admission Test) Quantitative Reasoning: Algebra syllabus?
GMAT (Graduate Management Admission Test) Quantitative Reasoning: Algebra is split into 5 chapters: Algebraic Expressions & Manipulation, Linear Equations & Inequalities, Quadratics & Higher-Order Equations, Functions, Formulas & Sequences and Coordinate Geometry (Algebra-based). Together they contain 25 topics and 10 sub-topics.
How many chapters are in GMAT (Graduate Management Admission Test) Quantitative Reasoning: Algebra?
5 chapters, containing 25 topics and 10 sub-topics. The largest chapter is Algebraic Expressions & Manipulation, with 5 topics.
Are there flashcards for GMAT (Graduate Management Admission Test) Quantitative Reasoning: Algebra?
Yes — a 50-card Quantitative Reasoning: Algebra deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.