Number of integral values of $a$ for which equation ${log}_{e} (x^{2} + 5 x) = {log}_{e} (x + a + 3)$ has exactly one solution is

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Solution

The correct option is B 5
$x^{2} + 5 x > 0 \Rightarrow x \in (- \infty, - 5) \cup (0, \infty)$
Now, $x^{2} + 5 x = x + a + 3$
$x^{2} + 4 x = a + 3$

So $0 < a + 3 \leq 5$
$- 3 < a \leq 2$
So $a = - 2, - 1, 0, 1, 2$

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