Question:medium

The number of integer solutions of the equation (x2−10)(x2−3x−10)=1 is

Updated On: Jan 15, 2026
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Correct Answer: 4

Solution and Explanation

Case 1: When \(x^2 - 3x - 10 = 0\) and \(x^2 - 10 e 0\)

Solve the quadratic equation: \[ x^2 - 3x - 10 = 0 \] Factorizing gives: \[ (x - 5)(x + 2) = 0 \] Thus, \(x = 5\) or \(x = -2\)

Verify that neither solution satisfies \(x^2 - 10 = 0\). Both are valid.

Case 2: When \(x^2 - 10 = 1\)

This leads to: \[ x^2 = 11 \] This equation has no integer solutions. Consequently, no values of \(x\) arise from this case.

Case 3: When \(x^2 - 10 = -1\) and \(x^2 - 3x - 10\) is an even number

This implies: \[ x^2 = 9 \] Therefore: \[ (x + 3)(x - 3) = 0 \Rightarrow x = -3 \text{ or } x = 3 \]

Check if \(x^2 - 3x - 10\) is even for these values:

  • For \(x = -3\): \(x^2 - 3x - 10 = 9 + 9 - 10 = 8\), which is even ✅
  • For \(x = 3\): \(x^2 - 3x - 10 = 9 - 9 - 10 = -10\), which is even ✅

Both values are valid.

Conclusion: The valid integer solutions for \(x\) are: \[ x = -3, -2, 3, 5 \] A total of 4 values satisfy the conditions.

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