Mathematics · Quadratic Equations

JEE Main 2025 — 4 April, Morning Shift — Question 30

Consider the equation $x^{2}+4 x-n=0$ where $n \in[20$ , 100] is a natural number. Then the number of all distinct values of $n$ , for which the given equation has integral roots, is equal to

Option A:
8
Option B:
6
Correct
Option C:
7
Option D:
5

Answer: B

Step-by-step solution

$x^{2}+4 x-n=0$ has integer roots

$\Rightarrow x=\frac{-4 \pm \sqrt{16+4 n}}{2}=-2 \pm \sqrt{4+n}$

For $x$ to be integer $4+n$ must be perfect squares

$n \in[20,100]$

$n+4 \in[24,104]=S$

$\left\{25,36, \ldots 10^{2}\right\} \in S \Rightarrow 5^{2}, 6^{2}, \ldots 10^{2} \Rightarrow 6$ values of $n$

Answer key and solution verified before publishing.

Practise Quadratic Equations

Start with this question, then two more from the same chapter — with a tutor that explains every step. Free.

Solve a similar one free

Exam: JEE Main 2025
Paper: 4 April, Morning Shift
Subject: Mathematics
Chapter: Quadratic Equations
Topic: Nature of Roots of Quadratic Equation