# What is the discriminant of x^2 - 3x - 5 and what does that mean?

Jul 18, 2015

Its discriminant $\Delta$ is $29$, which means that ${x}^{2} - 3 x - 5 = 0$ has two distinct real solutions. Since $29$ is not a perfect square, those solutions are irrational.

#### Explanation:

${x}^{2} - 3 x - 5$ is of the form $a {x}^{2} + b x + c$, with $a = 1$, $b = - 3$ and $c = - 5$.

Its discriminant $\Delta$ is given by the formula:

$\Delta = {b}^{2} - 4 a c = {\left(- 3\right)}^{2} - \left(4 \times 1 \times - 5\right) = 9 + 20 = 29$

Since $\Delta > 0$ the equation has two distinct real roots, but since $29$ is not a perfect square, those roots are irrational. That is they are not expressible as $\frac{p}{q}$ for some integers $p$ and $q$.

The solutions of ${x}^{2} - 3 x - 5 = 0$ are given by the quadratic formula:

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

$= \frac{3 \pm \sqrt{29}}{2}$

Notice that the discriminant $\Delta$ is the part under the square root. Hence if $\Delta > 0$ we get the two distinct real roots. If $\Delta = 0$ we get one repeated real root. If $\Delta < 0$ then the equation has no real roots (it has two distinct complex roots).