The number of distinct real solutions of the equation x x+4 +3 x+2+10=0 is [2026]

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Solution

Q.
The number of distinct real solutions of the equation $x$ $| x + 4 |$ $+ 3$ $| x + 2 |$ $+ 10 = 0$ is [2026]

1 1

2 2

3 3

4 0

Ans.
(1)

$Case I x < - 4$

$x (- x + 4) + 3 (- x + 2) + 10 = 0$

$x^{2} + 7 x - 4 = 0$

$\Rightarrow x = - \frac{7 + \sqrt{65}}{2} or - \frac{7 - \sqrt{65}}{2}$

$Reject$ $Accept$

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

$x (x + 4) + 3 (- (x + 2)) + 10 = 0$

$x^{2} + x + 4 = 0$

$D < 0 No solution$

$Case III x \geq - 1$

$x (x + 4) + 3 (x + 2) + 10 = 0$

$x^{2} + 7 x + 16 = 0$

$D < 0 No solution$

$\Rightarrow No. of solutions = 1$

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