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JEE Main 20254 AprilMorning Shift
Exam
JEE Main
Year
2025
Date
4 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Quadratic Equation

JEE Main 2025 · 4 April · Morning Shift · Mathematics · Quadratic Equation

Consider the equation x^2+4 x-n=0 , where n ∈[20,100] is a natural...

Consider the equation x2+4x−n=0\mathrm{x}^2+4 \mathrm{x}-\mathrm{n}=0, where n∈[20,100]\mathrm{n} \in[20,100] is a natural number. Then the number of all distinct values of nn, for which the given equation has integral roots, is equal to

Options

  1. A7
  2. B8
  3. C6
  4. D5

Correct answer

C. 6

Step-by-step solution

<br/>x2+4x+4=n+4(x+2)2=n+4x=−2±n+4∵20≤n≤10024≤n+4≤104⇒n+4∈{5,6,7,8,9,10}<br/>\begin{aligned}<br/>& \mathrm{x}^2+4 \mathrm{x}+4=\mathrm{n}+4 \\ & (\mathrm{x}+2)^2=\mathrm{n}+4 \\ & \mathrm{x}=-2 \pm \sqrt{\mathrm{n}+4} \\ & \because 20 \leq \mathrm{n} \leq 100 \\ & \sqrt{24} \leq \sqrt{\mathrm{n}+4} \leq \sqrt{104} \\ & \Rightarrow \sqrt{\mathrm{n}+4} \in\{5,6,7,8,9,10\}<br/>\end{aligned}
∴′6\therefore ' 6 ' integral values of ' nn ' are possible
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