GMAT Quantitative Reasoning Syllabus

State Vieta's formulas relating the roots $r_1,r_2$ of $ax^{2}+bx+c=0$ to its coefficients.

Sum of roots $r_1+r_2=-\frac{b}{a}$; product of roots $r_1 r_2=\frac{c}{a}$.

List the three common special-product/factoring identities.

$(a+b)^{2}=a^{2}+2ab+b^{2}$; $(a-b)^{2}=a^{2}-2ab+b^{2}$; $a^{2}-b^{2}=(a+b)(a-b)$.

What happens to an inequality when you multiply or divide both sides by a negative number?

The direction of the inequality sign must be reversed, e.g. $-2x<6$ becomes $x>-3$.

How do you solve the absolute-value inequality $|x|<a$ and $|x|>a$ for $a>0$?

$|x|<a$ means $-a<x<a$ (a single interval); $|x|>a$ means $x<-a$ or $x>a$ (two rays).

What is a function, and what are its domain and range?

A function assigns each input exactly one output. The domain is the set of allowed inputs; the range is the set of resulting outputs.

How do you compute a composite function $(f\circ g)(x)$?

Apply $g$ first, then $f$: $(f\circ g)(x)=f(g(x))$.

What condition must a function satisfy to have an inverse, and how do the graphs of $f$ and $f^{-1}$ relate?

It must be one-to-one (pass the horizontal-line test). The graph of $f^{-1}$ is the reflection of $f$ across the line $y=x$.

State the exponent rules for products, quotients, and powers of powers.

$a^{m}\cdot a^{n}=a^{m+n}$; $\frac{a^{m}}{a^{n}}=a^{m-n}$; $(a^{m})^{n}=a^{mn}$; and $a^{0}=1$, $a^{-n}=\frac{1}{a^{n}}$.

What is the sum of the interior angles of a triangle, and of an $n$-sided convex polygon?

A triangle's interior angles sum to $180^{\circ}$; an $n$-gon's sum to $(n-2)\times180^{\circ}$.

State the Pythagorean theorem and name the two most common integer right-triangle side ratios.

For legs $a,b$ and hypotenuse $c$: $a^{2}+b^{2}=c^{2}$. Common triples: $3\text{-}4\text{-}5$ and $5\text{-}12\text{-}13$.

Give the side ratios for a $45^{\circ}\text{-}45^{\circ}\text{-}90^{\circ}$ and a $30^{\circ}\text{-}60^{\circ}\text{-}90^{\circ}$ triangle.

$45\text{-}45\text{-}90$: sides in ratio $1:1:\sqrt{2}$. $30\text{-}60\text{-}90$: sides in ratio $1:\sqrt{3}:2$ (opposite $30^{\circ}$, $60^{\circ}$, $90^{\circ}$).

What does it mean for two triangles to be similar, and how do their areas compare?

Similar triangles have equal corresponding angles and proportional sides (ratio $k$). Their areas are in ratio $k^{2}$.

State the area formulas for a triangle, rectangle, and trapezoid.

Triangle: $A=\frac{1}{2}bh$. Rectangle: $A=lw$. Trapezoid: $A=\frac{1}{2}(b_1+b_2)h$.

State the formulas for the circumference and area of a circle of radius $r$.

Circumference $C=2\pi r$ (or $\pi d$); area $A=\pi r^{2}$.

How do you find the arc length and sector area for a central angle of $\theta$ degrees in a circle of radius $r$?

Arc length $=\frac{\theta}{360}\times2\pi r$; sector area $=\frac{\theta}{360}\times\pi r^{2}$.

State the volume and surface area of a rectangular box (cuboid) with dimensions $l,w,h$.

Volume $V=lwh$; surface area $S=2(lw+lh+wh)$.

State the volume and surface area formulas for a cylinder of radius $r$ and height $h$.

Volume $V=\pi r^{2}h$; total surface area $S=2\pi r^{2}+2\pi r h$.

State the volume formulas for a sphere and a cone of radius $r$ (cone height $h$).

Sphere: $V=\frac{4}{3}\pi r^{3}$ (surface $4\pi r^{2}$). Cone: $V=\frac{1}{3}\pi r^{2}h$.

State the distance and midpoint formulas for points $(x_1,y_1)$ and $(x_2,y_2)$.

Distance $d=\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}$; midpoint $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

What is the slope of a line through $(x_1,y_1)$ and $(x_2,y_2)$, and what are the slope conditions for parallel and perpendicular lines?

$m=\frac{y_2-y_1}{x_2-x_1}$. Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals, $m_1 m_2=-1$.

Give the slope-intercept and point-slope forms of a line's equation.

Slope-intercept: $y=mx+b$ ($b$ = $y$-intercept). Point-slope: $y-y_1=m(x-x_1)$.

How do you compute the arithmetic mean and the median of a data set?

Mean $=\frac{\text{sum of values}}{\text{number of values}}$. Median = middle value when data are ordered (average of the two middle values if the count is even).

Distinguish the mean, median, and mode as measures of central tendency.

Mean is the average (sensitive to outliers); median is the positional middle (resistant to outliers); mode is the most frequently occurring value.

What do range and standard deviation each measure, and how does standard deviation relate to variance?

Range = max $-$ min. Standard deviation measures spread about the mean; it is the square root of the variance ($\sigma=\sqrt{\sigma^{2}}$). Larger values mean more dispersion.

State the basic probability of an event and the rule for its complement.

$P(E)=\frac{\text{favorable outcomes}}{\text{total equally likely outcomes}}$, with $0\le P(E)\le1$. Complement: $P(\text{not }E)=1-P(E)$.

State the addition rule for the probability of $A$ or $B$, and simplify it for mutually exclusive events.

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$. If $A$ and $B$ are mutually exclusive, $P(A\cap B)=0$, so $P(A\cup B)=P(A)+P(B)$.

State the multiplication rule for two independent events $A$ and $B$.

For independent events, $P(A\cap B)=P(A)\times P(B)$.

How do permutations and combinations differ, and what are their formulas?

Permutations count ordered selections: $P(n,r)=\frac{n!}{(n-r)!}$. Combinations count unordered selections: $C(n,r)=\binom{n}{r}=\frac{n!}{r!\,(n-r)!}$.

When reading a pie chart versus a bar or line graph, what does each best represent?

A pie chart shows parts of a whole (proportions summing to $100\%$); a bar graph compares discrete categories; a line graph shows trends or change over a continuous variable such as time.

In data interpretation, how do you find the value of a slice from a pie chart given the total?

Multiply the total by the slice's percentage (or by $\frac{\text{central angle}}{360^{\circ}}$), e.g. a $25\%$ slice of a total of $800$ equals $0.25\times800=200$.