JEE Mains Maths · Quadratic Equations

Discriminant and Location of Roots

What the discriminant says about the roots — real, equal, rational or absent — and how the sign of the quadratic at chosen points places the roots on the number line.

All Quadratic Equations notes JEE Mains Maths strategy

Why this matters

Eighteen PYQs, thirteen of them multiple choice, and three from 2026. Nine use the sign of the discriminant — equal roots, real roots, no real roots, or a quadratic that keeps one sign for every x; three need the discriminant to be a perfect square, so the roots are rational or integers; six place the roots against given numbers — both positive, both negative, inside an interval, or on either side of a point. Three ideas cover the page.

Concept 1 of 3: The sign of the discriminant

D=b^2-4ac

decides the roots: two real roots when

D>0

, one repeated root when

D=0

, none when

D<0

. The same number controls the sign of the quadratic: with

a>0

and

D<0

the graph never reaches the axis, so the quadratic is positive for every

x

. When

x

and

y

are tied by one equation, read it as a quadratic in one of them; for that one to be real, its discriminant must be

\ge0

Definition

$D>0$ : two distinct real roots; $D=0$ : equal roots; $D<0$ : no real roots.
$ax^2+bx+c>0$ for all $x$ exactly when $a>0$ and $D<0$ .
$ax^2+bx+c<0$ for all $x$ exactly when $a<0$ and $D<0$ .
If the $x^2$ coefficient can be $0$ , check that value on its own.

Discriminant

D=b^2-4ac,\qquad x=\frac{-b\pm\sqrt D}{2a}

Worked example

For which

m

does

x^2-2mx+4m-3=0

have equal roots?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 3 Apr 2025 · Q132Moderate

Example 1 · Quadratic Equations · Discriminant and Location of Roots

Let the equation

x(x+ 2)(12 -k) = 2

have equal roots. Then the distance of the point

\left( k,\frac{k}{2} \right)

from the line

3x+ 4y+ 5 = 0

is.

The x² coefficient can vanish

When the leading coefficient holds a parameter, the value that makes it

0

leaves a linear equation, and the discriminant test does not apply. Treat that value on its own — it is often the case the options exclude.

Drill 8 more on the sign of the discriminant

Concept 2 of 3: Rational and integer roots

With integer coefficients, the roots are rational exactly when

D

is a perfect square: the square root in the formula disappears. For

x^2+bx+c=0

with integers

b,c

, rational roots are integers. So the question becomes one of two searches: for which parameters is

D

a perfect square, or which factor pairs of

c

add up to

-b

Definition

Integer coefficients: roots rational exactly when $D$ is a perfect square.
$x^2+bx+c=0$ , $b,c$ integers: rational roots are integers.
Integer roots: list the factor pairs of $c$ whose sum is $-b$ .

Rational roots

\sqrt{b^2-4ac}\in\mathbb{Z}\ \Rightarrow\ x=\frac{-b\pm\sqrt{D}}{2a}\ \text{is rational}

Worked example

The roots of

x^2-px+12=0

are integers. How many values can

p

take?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 4 April 2025 · Q68Moderate

Example 2 · Quadratic Equations · Discriminant and Location of Roots

Consider the equation

x^{2}+ 4x-n= 0

, where

n \in \lbrack 20,100\rbrack

is a natural number. Then the number of all distinct values of

n

, for which the given equation has integral roots, is equal to.

Rational coefficients first

The perfect-square test works only when the coefficients are rational. With

\sqrt2

in a coefficient, the roots can be irrational even when

D

is a perfect square.

Drill 2 more on rational and integer roots

Concept 3 of 3: Placing the roots on the number line

For

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

with

a>0

, the graph is a U. A number

k

lies between the roots exactly when

f(k)<0

— no discriminant check is needed. For both roots beyond

k

on the same side, three conditions are needed: real roots, the vertex on that side, and

f(k)>0

. For both roots positive, the sum and product do the same job. For one root in each of two intervals, check the sign of

f

at the end points.

Definition

$k$ between the roots: $af(k)<0$ .
Both roots $>k$ : $D\ge0$ , $-\frac{b}{2a}>k$ , $af(k)>0$ .
Both roots positive: $D\ge0$ , sum $>0$ , product $>0$ .
Both roots in $(p,q)$ : $D\ge0$ , $p<-\frac{b}{2a}<q$ , $af(p)>0$ , $af(q)>0$ .

A number between the roots

\alpha<k<\beta\quad\text{exactly when}\quad a\,f(k)<0

Worked example

For which

m

are both roots of

x^2-2mx+m+6=0

positive?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 4 Apr 2026 Shift 2 · Q55Moderate

Example 3 · Quadratic Equations · Discriminant and Location of Roots

If the quadratic equation

(\lambda+ 2)x^{2}- 3\lambda x+ 4\lambda= 0

\lambda\neq - 2

, has two positive roots, then the number of possible integral values of

\lambda

is :

The vertex condition is not optional

D\ge0

and

f(k)>0

hold both when the two roots are above

k

and when both are below it. Only the vertex

-\frac{b}{2a}

tells which side.

Drill 5 more on placing the roots on the number line

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

The sign of the discriminant
Discriminant
$D=b^2-4ac,\qquad x=\frac{-b\pm\sqrt D}{2a}$
Rational and integer roots
Rational roots
$\sqrt{b^2-4ac}\in\mathbb{Z}\ \Rightarrow\ x=\frac{-b\pm\sqrt{D}}{2a}\ \text{is rational}$
Placing the roots on the number line
A number between the roots
$\alpha<k<\beta\quad\text{exactly when}\quad a\,f(k)<0$

Watch out for (3)

The x² coefficient can vanish
→ The sign of the discriminant
Rational coefficients first
→ Rational and integer roots
The vertex condition is not optional
→ Placing the roots on the number line

Test yourself on Quadratic Equations

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Quadratic Equations chapter test Drill all 18 questions