Question

The number of distinct real solutions of the equation \[ x|x + 4| + 3|x + 2| + 10 = 0 \] is

• A. 2
• B. 0
• C. 3
• D. 1

Hint:

For equations involving absolute values, always split the number line using points where expressions inside absolute values become zero, then solve case by case.

Answer 1

The Correct Option is A

The given equation involves absolute value expressions. We solve it by considering different intervals based on the critical points \[ x = -4 \quad \text{and} \quad x = -2. \] Step 1: Case I — \( x \ge -2 \).
For \( x \ge -2 \), \[ |x + 4| = x + 4,\quad |x + 2| = x + 2. \] Substituting, \[ x(x + 4) + 3(x + 2) + 10 = 0. \] \[ x^2 + 4x + 3x + 6 + 10 = 0 \Rightarrow x^2 + 7x + 16 = 0. \] Discriminant: \[ \Delta = 49 - 64 = -15<0. \] So, no real solution in this interval.
Step 2: Case II — \( -4 \le x<-2 \).
For \( -4 \le x<-2 \), \[ |x + 4| = x + 4,\quad |x + 2| = -(x + 2). \] Substituting, \[ x(x + 4) + 3(-x - 2) + 10 = 0. \] \[ x^2 + 4x - 3x - 6 + 10 = 0 \Rightarrow x^2 + x + 4 = 0. \] Discriminant: \[ \Delta = 1 - 16 = -15<0. \] So, no real solution in this interval.
Step 3: Case III — \( x<-4 \).
For \( x<-4 \), \[ |x + 4| = -(x + 4),\quad |x + 2| = -(x + 2). \] Substituting, \[ x(-x - 4) + 3(-x - 2) + 10 = 0. \] \[ - x^2 - 4x - 3x - 6 + 10 = 0 \Rightarrow -x^2 - 7x + 4 = 0. \] Multiplying by \( -1 \), \[ x^2 + 7x - 4 = 0. \] \[ x = \frac{-7 \pm \sqrt{49 + 16}}{2} = \frac{-7 \pm \sqrt{65}}{2}. \] Both roots satisfy \( x<-4 \), hence both are valid.
Step 4: Count the solutions.
There are exactly two distinct real solutions.
Final Answer: \[ \boxed{2} \]