# What are the value of k, for which x^2+5kx+16=0 has no real roots?

Nov 2, 2016

The values of k for which the equation ${x}^{2} + 5 k x + 16 = 0$ has no real roots are $- \frac{8}{5} < k < \frac{8}{5}$

#### Explanation:

An equation $a {x}^{2} + b x + c = 0$ has no real roots if the determinant ${b}^{2} - 4 a c < 0$

${x}^{2} + 5 k x + 16 = 0$ has no real roots if the determinant ${\left(5 k\right)}^{2} - 4 \times 1 \times 16 = 25 {k}^{2} - 64 < 0$ or

$\left(5 k - 8\right) \left(5 k + 8\right) < 0$

As $\left(5 k - 8\right) \left(5 k + 8\right)$ is negative,

either $5 k + 8 < 0$ and $5 k - 8 > 0$

But this is just not possible, as $k$ cannot be $k < - \frac{8}{5}$ as well as $k > \frac{8}{5}$

Other alternative is $5 k + 8 > 0$ and $5 k - 8 < 0$ i.e.

$k > - \frac{8}{5}$ as well as $k < \frac{8}{5}$. This is possible if value of $k$ is between $- \frac{8}{5}$ and $\frac{8}{5}$.

Hence $- \frac{8}{5} < k < \frac{8}{5}$