Question

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

• A. The equation has three solutions whose sum is 12.
• B. The equation has four solutions whose sum is 10.
• C. The equation has only two solutions, one positive and one negative.
• D. The equation has only two solutions, which are greater than 5.

Hint:

When solving equations with absolute values, consider both positive and negative cases for the expressions inside the absolute value.

Answer 1

The Correct Option is B

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).