Question:

Which of the following statements is true about the solutions of the equation \(\lvert x^2 - 5x \rvert = 6\)?

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When solving equations with absolute values, consider both positive and negative cases for the expressions inside the absolute value.
Updated On: Apr 19, 2025
  • The equation has three solutions whose sum is 12.
  • The equation has four solutions whose sum is 10.
  • The equation has only two solutions, one positive and one negative.
  • The equation has only two solutions, which are greater than 5.
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The Correct Option is B

Solution and Explanation

The given equation is \(\lvert x^2 - 5x \rvert = 6\). We consider the two cases for the absolute value function: 1. \(x^2 - 5x = 6\) 2. \(x^2 - 5x = -6\) For the first case: \[ x^2 - 5x - 6 = 0 \] Factoring: \[ (x - 6)(x + 1) = 0 \] So, \(x = 6\) or \(x = -1\). For the second case: \[ x^2 - 5x + 6 = 0 \] Factoring: \[ (x - 3)(x - 2) = 0 \] So, \(x = 3\) or \(x = 2\). Thus, the four solutions are \(x = 6, -1, 3, 2\), and their sum is \(6 + (-1) + 3 + 2 = 10\). Thus, the correct answer is option (2).
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