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

1 Answer
Oct 11, 2015

Answer:

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.