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Quadratic Equation and Inequalities question

2024 · 30 Jan · Shift 2 · Q53
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Quadratic Equation and Inequalities question

2024 · 30 Jan · Shift 2 · Q53

JEE MainMathematicsQuadratic Equation and InequalitiesNumerical+4 / −1
The number of real solutions of the equation x(x2+3∣x∣+5∣x−1∣+6∣x−2∣)=0x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0x(x2+3∣x∣+5∣x−1∣+6∣x−2∣)=0 is ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 1

  1. We need the number of real solutions of x(x2+3∣x∣+5∣x−1∣+6∣x−2∣)=0.x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0.x(x2+3∣x∣+5∣x−1∣+6∣x−2∣)=0.

  2. A product is zero if at least one factor is zero. So either:

    • x=0x=0x=0, or
    • x2+3∣x∣+5∣x−1∣+6∣x−2∣=0.x^2+3|x|+5|x-1|+6|x-2|=0.x2+3∣x∣+5∣x−1∣+6∣x−2∣=0.

  3. Now examine the second factor: x2+3∣x∣+5∣x−1∣+6∣x−2∣.x^2+3|x|+5|x-1|+6|x-2|.x2+3∣x∣+5∣x−1∣+6∣x−2∣. Each term is non-negative for every real xxx:

  • x2≥0x^2 \ge 0x2≥0
  • 3∣x∣≥03|x| \ge 03∣x∣≥0
  • 5∣x−1∣≥05|x-1| \ge 05∣x−1∣≥0
  • 6∣x−2∣≥06|x-2| \ge 06∣x−2∣≥0

Hence their sum is always ≥0\ge 0≥0.

  1. For the sum to be equal to 000, every non-negative term must individually be zero. So we would need: x2=0,∣x∣=0,∣x−1∣=0,∣x−2∣=0.x^2=0, \quad |x|=0, \quad |x-1|=0, \quad |x-2|=0.x2=0,∣x∣=0,∣x−1∣=0,∣x−2∣=0. These imply simultaneously: x=0,x=0,x=1,x=2,x=0,\quad x=0,\quad x=1,\quad x=2,x=0,x=0,x=1,x=2, which is impossible.

So, x2+3∣x∣+5∣x−1∣+6∣x−2∣>0x^2+3|x|+5|x-1|+6|x-2|>0x2+3∣x∣+5∣x−1∣+6∣x−2∣>0 for every real xxx.

  1. Therefore the only way the product can be zero is: x=0.x=0.x=0.

  2. Hence the equation has exactly one real solution.

Therefore, the number of real solutions is 1.\boxed{1}.1​.

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