Question

The number of natural numbers $n$ for which the equation $\left(x - 8\right)x=n\left(n - 10\right)$ has no real solution is equal to

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

Answer: (D)

$x^{2}-8x-n\left(n - 10\right)=0$ has no real solution.
$\Rightarrow 64+4n\left(n - 10\right) < 0$
$\Rightarrow n^{2}-10n+16 < 0$
$\Rightarrow \left(n - 8\right)\left(n - 2\right) < 0$
$\Rightarrow 2 < n < 8$
$\Rightarrow n=3,4,5,6,7$
$\Rightarrow $ Number of values of $n$ is $5$