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JEE Main 20257 AprilEvening Shift
Exam
JEE Main
Year
2025
Date
7 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Quadratic Equation

JEE Main 2025 · 7 April · Evening Shift · Mathematics · Quadratic Equation

The number of real roots of the equation x|x-2|+3|x-3|+1=0 is :

The number of real roots of the equation x∣x−2∣+3∣x−3∣+1=0\mathrm{x}|\mathrm{x}-2|+3|\mathrm{x}-3|+1=0 is :

Options

  1. A4
  2. B2
  3. C1
  4. D3

Correct answer

C. 1

Step-by-step solution

 (I) x<2−x2+2x−3x+9+1=0⇒x2+x−10=0⇒x=−1+412,−1−412×\begin{aligned}& \text{ (I) } x < 2 \\ & -x^2+2 x-3 x+9+1=0 \\ & \Rightarrow x^2+x-10=0 \\ & \Rightarrow x=\frac{-1+\sqrt{41}}{2}, \frac{-1-\sqrt{41}}{2} \\ & \quad \times \quad \sqrt{ }\end{aligned} (II) 2≤x<3⇒x2−2x−3x+9+1=0⇒x2−5x+10=0 D <0 (III) x≥3x2−2x+3x−9+2=0⇒x2+x−8=0x=−1+322,−1−322××\begin{aligned}& \text{ (II) } 2 \leq x < 3 \\ & \Rightarrow x^2-2 x-3 x+9+1=0 \\ & \Rightarrow x^2-5 x+10=0 \\ & \text{ D } < 0 \\ & \text{ (III) } x \geq 3 \\ & x^2-2 x+3 x-9+2=0 \\ & \Rightarrow x^2+x-8=0 \\ & x=\frac{-1+\sqrt{32}}{2}, \frac{-1-\sqrt{32}}{2} \\ & \quad \times \quad \times\end{aligned}$1 real roots
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