Question

How do you use the discriminant to find the number of real solutions of the following quadratic equation: `2x^2 + 4x + 2 = 0`?

Answer 1

`b^2-4ac = 4^2 - 4(2)(2) = 0` so this equation has exactly one real solution.

Explanation:

For a quadratic of the form `ax^2+bx+c = 0`
the solution roots are given by the quadratic formula: `x=(-b+-sqrt(b^2-4ac))/(2a)`

The discriminant is the portion `(b^2-4ac)`
if it is negative then the roots include a term which is the square root of a negative number; since there are no Real numbers whose square root is negative, when the discriminant is `<0` there are no Real roots.

If the discriminant is `=0` then `+-sqrt(0)` is a single value (`+0 = -0`)
and the quadratic has exactly one Real solution.

If the discriminant is `>0` then `+-sqrt("discriminant")` represents two different values and the quadratic has two Real solutions.