# What is the discriminant of x^2 + 25 = 0 and what does that mean?

Jul 18, 2015

${x}^{2} + 25 = 0$ has discriminant $- 100 = - {10}^{2}$

Since this is negative the equation has no real roots. Since it is negative of a perfect square it has rational complex roots.

#### Explanation:

${x}^{2} + 25$ is in the form $a {x}^{2} + b x + c$, with $a = 1$, $b = 0$ and $c = 25$.

This has discriminant $\Delta$ given by the formula:

$\Delta = {b}^{2} - 4 a c = {0}^{2} - \left(4 \times 1 \times 25\right) = - 100 = - {10}^{2}$

Since $\Delta < 0$ the equation ${x}^{2} + 25 = 0$ has no real roots. It has a pair of distinct complex conjugate roots, namely $\pm 5 i$

The discriminant $\Delta$ is the part under the square root in the quadratic formula for roots of $a {x}^{2} + b x + c = 0$ ...

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a} = \frac{- b \pm \sqrt{\Delta}}{2 a}$

So if $\Delta > 0$ the equation has two distinct real roots.

If $\Delta = 0$ the equation has one repeated real root.

If $\Delta < 0$ the equation has no real roots, but two distinct complex roots.

In our case the formula gives:

$x = \frac{- 0 \pm 10 i}{2} = \pm 5 i$