🔢 Quant

Quadratics, Functions & Graphs

Factoring identities, the discriminant, vertex max/min, function notation, and lines in the coordinate plane.

~6h

to master

Quadratics: factor first, formula last

A quadratic equation in standard form is $ax^2 + bx + c = 0$ with $a \neq 0$ . The GMAT's favourite solving route is factoring — move everything to one side, factor, and use the zero-product rule: a product is 0 exactly when some factor is 0.

📐Core Rule

The three identities that unlock most factoring: $a^2 - b^2 = (a+b)(a-b)$ · $a^2 + 2ab + b^2 = (a+b)^2$ · $a^2 - 2ab + b^2 = (a-b)^2$ . Spot them in disguise: $x^2 - 49$ , $4y^2 - 12y + 9 = (2y-3)^2$ , $\frac{x^2 - 1}{x - 1} = x + 1$ (for $x \neq 1$ ).

When factoring stalls, the quadratic formula always works: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The discriminant $b^2 - 4ac$ tells you how many real roots exist without solving:

$b^2 - 4ac$	Real roots
positive	two distinct
zero	exactly one
negative	none

Count roots, don't find them: .

⚠️GMAT Trap

Never divide both sides by a variable expression. $x^2 = 5x$ → dividing by $x$ silently discards the solution $x = 0$ . Factor instead: $x(x - 5) = 0$ , so $x = 0$ or $x = 5$ . The "lost root" is a stock wrong answer.

Max/min: the vertex in one line

For

f(x) = ax^2 + bx + c

: if

a > 0

the parabola opens up (unique minimum); if

a < 0

it opens down (unique maximum). The extreme value occurs at

x = -\dfrac{b}{2a}

and equals

c - \dfrac{b^2}{4a}

. Revenue/area/projectile problems are this formula wearing a costume — try .

Functions: machine in, machine out

$f(x) = x^2 - 3x$ defines a machine: input $x$ , output $f(x)$ . Evaluate by substitution everywhere the variable appears: $f(-2) = 4 + 6 = 10$ .

Domain = allowed inputs (exclude division by zero, negative numbers under square roots).
Range = achievable outputs (a quadratic's range starts/ends at its vertex value).
One input gives at most one output, but different inputs may share an output.
For compound machines, work inside-out: $f(g(2))$ means run $g$ first.

Graphs and lines

Every non-vertical line is $y = mx + b$ : slope $m$ , $y$ -intercept $b$ .

📐Core Rule

$m = \dfrac{y_2 - y_1}{x_2 - x_1}$ — subtract coordinates in the same order. Positive slope rises left-to-right, negative falls, zero is horizontal; vertical lines ( $x = c$ ) have undefined slope. Parallel lines share $m$ ; perpendicular slopes multiply to $-1$ .

$x$ -intercept: set $y = 0$ . $y$ -intercept: set $x = 0$ . For any function, the $x$ -intercepts are the solutions of $f(x) = 0$ — algebra and geometry are the same fact.
Two distinct lines intersect at the unique shared solution of their equations; parallel ⇔ no solution; same line ⇔ infinitely many. This is the graphical face of the linear-systems trichotomy.

💡Exam Tip

A point lies on a graph exactly when its coordinates satisfy the equation — checking membership is substitution, never plotting. For "which quadrant" questions, just track the two signs: $(+,-)$ → IV.

Checklist

One side $= 0$ before factoring; never divide away a variable
Discriminant for how many roots; formula for which
Vertex $x = -b/2a$ for any max/min wording
Domain: ban zero denominators and negative radicands
Slope subtraction in consistent order

Sample Questions

22 practice questions

Medium

For which value of $k$ does the equation $x^2 - 6x + k = 0$ have exactly one real solution?

Hard

A farmer will use 40 meters of fencing to enclose a rectangular pen on three sides, with an existing wall forming the fourth side. What is the greatest possible area of the pen, in square meters? (Note: The area of a rectangle equals its length multiplied by its width.)

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