# How do you use the discriminant to determine the nature of the roots for 4x^2 + 15x + 10 = 0?

Jun 27, 2015

$4 {x}^{2} + 15 x + 10$ has discriminant $\Delta = 65$

So $4 {x}^{2} + 15 x + 10 = 0$ has two distinct real, irrational roots.

#### Explanation:

$4 {x}^{2} + 15 x + 10$ is of the form $a {x}^{2} + b x + c$
with $a = 4$, $b = 15$ and $c = 10$.

The discriminant is given by the formula:

$\Delta = {b}^{2} - 4 a c = {15}^{2} - \left(4 \times 4 \times 10\right) = 225 - 160 = 65$

This is positive, but not a perfect square. So the quadratic equation has two distinct irrational real roots.

The various possible cases (assuming that the quadratic has rational coefficients) are as follows:

$\Delta > 0$ The equation has two distinct real roots. If $\Delta$ is a perfect square then the roots are rational too. Otherwise they are irrational.

$\Delta = 0$ The equation has one (repeated) rational real root.

$\Delta < 0$ The equation has no real roots. It has two distinct complex roots, which are complex conjugates of one another.