# A quadratic equation has a discriminant of 14. What is the number and type of solutions of the equation?

Oct 11, 2015

It has two distinct Real solutions.

If the coefficients of the quadratic equation are rational (e.g. integers), then we can be sure that those solutions will both be irrational.

#### Explanation:

A quadratic equation of the for $a {x}^{2} + b x + c = 0$ has discriminant $\Delta$ given by the formula:

$\Delta = {b}^{2} - 4 a c$

It has roots given by the quadratic formula:

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

The discriminant $\Delta$ is the part of the quadratic formula under the square root.

So if $\Delta < 0$, then $\sqrt{\Delta} = i \sqrt{- \Delta}$ is pure imaginary and the quadratic equation has two distinct Complex roots that are Complex conjugates of one another.

If $\Delta = 0$, then the quadratic equation has one (repeated) root. We call such a quadratic a perfect square trinomial.

If $\Delta > 0$ is a perfect square then the equation will have two distinct rational roots (assuming the coefficients are rational).

If $\Delta > 0$ is not a perfect square, then the equation will still have two distinct roots, but they will be irrational - unless the coefficients of the polynomial are themselves irrational and suitably chosen.