Question

If the set of all solutions of \(|x^2 + x - 9| = |x| + |x^2 - 9|\) is \([\alpha, \beta] \cup [\gamma, \infty)\), then \((\alpha^2 + \beta^2 + \gamma^2)\) is equal to:

• A. 9
• B. 18
• C. 36
• D. 72

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The equation is of the form \(|a + b| = |a| + |b|\). This equality holds true if and only if \(a \cdot b \geq 0\). Here, \(a = x\) and \(b = x^2 - 9\).

Step 2: Key Formula or Approach:
We must solve the inequality: \[ x(x^2 - 9) \geq 0 \] \[ x(x - 3)(x + 3) \geq 0 \]

Step 3: Detailed Explanation:
1. Find the critical points: \(x = -3, 0, 3\). 2. Test the intervals using the wavy curve method: - For \(x>3\): (+)(+)(+) \(\to\) Positive. - For \(0<x<3\): (+)(-)(+) \(\to\) Negative. - For \(-3<x<0\): (-)(-)(+) \(\to\) Positive. - For \(x<-3\): (-)(-)(-) \(\to\) Negative. 3. The solution set is \([-3, 0] \cup [3, \infty)\). 4. Comparing with \([\alpha, \beta] \cup [\gamma, \infty)\): - \(\alpha = -3, \beta = 0, \gamma = 3\). 5. Calculate \(\alpha^2 + \beta^2 + \gamma^2\): \[ (-3)^2 + 0^2 + 3^2 = 9 + 0 + 9 = 18 \]

Step 4: Final Answer:
The value of \((\alpha^2 + \beta^2 + \gamma^2)\) is 18.