🔢 Quant

Quadratic Equations

Roots, discriminant, and maxima/minima of quadratic expressions.

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

1/5

of recent CAT years

% of Quant📈

~0%

of section questions

Est. hours⏱️

~8h

to master

~1/22

2021

2022

2023

2024

2025

🎯PYQ Evidence

CAT 2021–2025: ~0.1 per slot (2021: 0.3). Only one question is tagged purely 'quadratic' (2021) — but quadratic technique (Vieta, discriminant, factoring) runs through the ~40 equation questions of 2021–2025. Learn it as a tool, not a topic.

Quadratic Equations

A quadratic $ax^2+bx+c=0$ is the most-tested single object in CAT algebra. You almost never need the full formula — Vieta's relations and the discriminant answer most questions directly.

The two tools

$\text{sum of roots}=\alpha+\beta=-\frac{b}{a},\qquad \text{product}=\alpha\beta=\frac{c}{a}.$

$\text{Discriminant}\ D=b^2-4ac:\quad D>0\ \text{(two real)},\ \ D=0\ \text{(equal)},\ \ D<0\ \text{(none real)}.$

Build a quadratic from its roots: $x^2-(\alpha+\beta)x+\alpha\beta=0$ .

A worked example

The roots of a quadratic satisfy $\alpha+\beta=5$ and $\alpha\beta=6$ . Find $\alpha^2+\beta^2$ .

Use the identity $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$ — no need to find the roots:

$\alpha^2+\beta^2=5^2-2(6)=25-12=\mathbf{13}.$

(The roots happen to be $2$ and $3$ , but Vieta got the answer without solving.)

Handy symmetric identities

$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$
$\dfrac1\alpha+\dfrac1\beta=\dfrac{\alpha+\beta}{\alpha\beta}$
$(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=\dfrac{D}{a^2}$

🎯PYQ Evidence

Complete the square first — the vertex form makes root-counting visible. : rewrite the quadratic as (x−2)² − 17 so its minimum of −17 is in plain sight, then read |f(x)| = r as the two equations f(x) = r and f(x) = −r. Each branch gives 0, 1, or 2 roots depending on r versus that range, and exactly three distinct roots can only happen when one branch is tangent (1 root) and the other cuts twice — which pins r = 17. Completing the square turns an opaque modulus equation into a clean count of cases.

Common traps

Sign of $b$ . Sum of roots is $-b/a$ — the minus sign trips people constantly.
"Real and distinct" vs "real". Distinct needs $D>0$ strictly; $D=0$ is real but repeated.
Maximum/minimum. The vertex value $-\dfrac{D}{4a}$ gives the extreme of $ax^2+bx+c$ .