Quadratic EquationMedium

Question

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

Options

A.3
B.1
C.0
D.2

Solution

Case I $x < - 4$

$${x( - (x + 4)) + 3( - (x + 2)) + 10 = 0 }{x^{2} + 7x - 4 = 0 }$$$\Rightarrow x = - \frac{7 + \sqrt{65}}{2}$ or $- \frac{7 - \sqrt{65}}{2}$

Reject Acept

Case II $- 4 \leq x < - 2$

$${x(x + 4) + 3( - (x + 2)) + 10 = 0 }{x^{2} + x + 4 = 0 }$$$D < 0$ No solution

Case III $x \geq - 1$

$${x(x + 4) + 3(x + 2) + 10 = 0 }{x^{2} + 7x + 16 = 0 }{D < 0 }$$No solution

⇒ No. of solution $= 1$.

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