Question

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

Answer 1

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 `ax^2+bx+c = 0` has discriminant `Delta` given by the formula:

`Delta = b^2 - 4ac`

It has roots given by the quadratic formula:

`x = (-b +-sqrt(b^2-4ac))/(2a) = (-b+-sqrt(Delta))/(2a)`

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.