GRE (Graduate Record Examinations) Quantitative Reasoning Syllabus

What is the distributive property and how is it used to expand a(b + c)?

a(b + c) = ab + ac. It distributes multiplication over addition/subtraction and is the basis for expanding and factoring expressions.

What does it mean to combine like terms in an algebraic expression?

Like terms have identical variable parts (same variables to the same powers). Combine them by adding/subtracting their coefficients, e.g. 3x + 5x = 8x; 3x and 3x² are not like terms.

Give the FOIL expansion of (a + b)(c + d).

ac + ad + bc + bd (First, Outer, Inner, Last).

State the three common special-product/difference factoring identities.

(a+b)² = a²+2ab+b²; (a−b)² = a²−2ab+b²; a²−b² = (a+b)(a−b).

How do you solve a linear equation in one variable such as 3x + 5 = 20?

Isolate the variable using inverse operations: subtract 5 (3x = 15), then divide by 3 (x = 5). Keep the equation balanced by doing the same to both sides.

What special rule applies when solving inequalities that does not apply to equations?

When multiplying or dividing both sides by a negative number, you must reverse the inequality sign (e.g. −2x < 6 becomes x > −3).

How do you solve a system of two linear equations by substitution or elimination?

Substitution: solve one equation for a variable, substitute into the other. Elimination: add/subtract multiples of the equations to cancel one variable. Then back-solve for the remaining variable.

What is the standard form of a quadratic equation and the quadratic formula?

Standard form: ax² + bx + c = 0 (a≠0). Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a).

What does the discriminant b² − 4ac tell you about a quadratic's roots?

Positive → two distinct real roots; zero → one repeated real root; negative → no real roots (two complex roots).

How do you solve a quadratic by factoring using the zero-product property?

Factor into (x − r)(x − s) = 0, then set each factor to zero, giving x = r or x = s. The zero-product property says if a product is 0, at least one factor is 0.

What defines a function, and how does the vertical line test identify one from a graph?

A function assigns exactly one output to each input. A graph represents a function if no vertical line crosses it more than once.

What are the domain and range of a function?

Domain = the set of all allowable input (x) values; Range = the set of all resulting output (y) values.

If f(x) = 2x² − 3, how do you evaluate f(−2)?

Substitute −2 for x: f(−2) = 2(−2)² − 3 = 2(4) − 3 = 8 − 3 = 5.

What is the slope formula and the slope-intercept form of a line?

Slope m = (y₂ − y₁)/(x₂ − x₁). Slope-intercept form: y = mx + b, where m is slope and b is the y-intercept.

What are the slope relationships for parallel and perpendicular lines?

Parallel lines have equal slopes (m₁ = m₂). Perpendicular lines have slopes that are negative reciprocals (m₁ × m₂ = −1).

State the distance formula and midpoint formula between two points.

Distance = √[(x₂−x₁)² + (y₂−y₁)²]. Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).

When translating a word problem, what does 'is', 'of', 'more than', and 'per' typically map to?

'is' → equals (=); 'of' → multiply (×); 'more than'/'sum' → add (+); 'per' → divide (rate). Identify the unknown and assign a variable first.

What are complementary and supplementary angles?

Complementary angles sum to 90°; supplementary angles sum to 180°.

What are vertical angles and what is true of angles formed when a transversal crosses parallel lines?

Vertical (opposite) angles are equal. With parallel lines and a transversal: corresponding and alternate (interior/exterior) angles are equal; co-interior (same-side) angles are supplementary.

What is the sum of the interior angles of any triangle, and the Triangle Inequality?

Interior angles sum to 180°. Triangle Inequality: the sum of any two side lengths must be greater than the third side.

State the Pythagorean theorem and name two common Pythagorean triples.

For a right triangle, a² + b² = c² (c is the hypotenuse). Common triples: 3-4-5 and 5-12-13 (and multiples).

What are the side ratios for 45-45-90 and 30-60-90 triangles?

45-45-90: sides 1 : 1 : √2. 30-60-90: sides 1 : √3 : 2 (opposite 30°, 60°, 90° respectively).

What is the area formula for a triangle, and what does similarity of triangles imply?

Area = ½ × base × height. Similar triangles have equal corresponding angles and proportional corresponding sides.

What is the sum of interior angles of an n-sided polygon, and the area of a parallelogram and trapezoid?

Interior angle sum = (n − 2) × 180°. Parallelogram area = base × height. Trapezoid area = ½ × (b₁ + b₂) × height.

State the circumference and area formulas of a circle, and the relation of an inscribed angle to its central angle.

Circumference = 2πr (= πd); Area = πr². An inscribed angle is half the central angle subtending the same arc.

What are the surface area and volume formulas for a rectangular solid and a cylinder?

Rectangular solid: V = lwh, SA = 2(lw + lh + wh). Cylinder: V = πr²h, lateral SA = 2πrh, total SA = 2πr² + 2πrh.

How do mean, median, and mode differ as measures of center?

Mean = sum ÷ count (sensitive to outliers). Median = middle value when ordered (or average of two middle values). Mode = most frequently occurring value.

What is the range, and how is standard deviation interpreted as a measure of spread?

Range = maximum − minimum. Standard deviation measures the typical distance of data values from the mean; a larger SD means more spread/dispersion.

When reading data from graphs, what is the difference between a frequency in a bar chart and a percentage in a pie chart?

A bar chart's height gives an absolute count/value; a pie chart's slice gives a proportion of the whole (multiply the percent by the total to get an absolute count).

What is the Fundamental Counting Principle, and the formulas for permutations and combinations?

If one event has m ways and another n ways, together there are m×n ways. Permutations (order matters): nPr = n!/(n−r)!. Combinations (order doesn't matter): nCr = n!/[r!(n−r)!].

What is the basic probability formula, and how do you combine probabilities for 'and' (independent) vs 'or' (mutually exclusive)?

P(event) = favorable outcomes ÷ total outcomes. Independent 'and': P(A and B) = P(A)×P(B). Mutually exclusive 'or': P(A or B) = P(A)+P(B). General 'or': P(A)+P(B)−P(A and B).

In a normal distribution, what percent of data falls within 1, 2, and 3 standard deviations of the mean (68-95-99.7 rule)?

About 68% within ±1 SD, about 95% within ±2 SD, and about 99.7% within ±3 SD of the mean.

In GRE Quantitative Comparison, what do the four answer choices mean and when is the answer 'D'?

A: Quantity A is greater; B: Quantity B is greater; C: the two are equal; D: the relationship cannot be determined from the information given. Choose D when the comparison changes for different valid values.