Question

What is the discriminant of `2x^2 + 5x + 5 = 0` and what does that mean?

Answer 1

For this quadratic, `Delta = -15`, which means that the equation has no real solutions, but it does have two distinct complex ones.

Explanation:

The general form for a quadratic equation is

`ax^2 + bx + c = 0`

The general form of the discriminant looks like this

`Delta = b^2 - 4 * a * c`

Your equation looks like this

`2x^2 + 5x + 5 = 0`

which means that you have

`{(a=2), (b=5), (c=5) :}`

The discriminant will thus be equal to

`Delta = 5^2 - 4 * 2 * 5`

`Delta = 25 - 40 = color(green)(-15)`

The two solutions for a general quadratic are

`x_(1,2) = (-b +- sqrt(Delta))/(2a)`

When `Delta<0`, such as you have here, the equation is said to have no real solutions, since you're extracting the square root from a negative number.

However, it does have two distinct complex solutions that have the general form

`x_(1,2) = (-b +- isqrt(-Delta))/(2a)`, when `Delta<0`

In your case, these solutions are

`x_(1,2) = (-5 +- sqrt(-15))/(4) = {(x_1 = (-5 + isqrt(15))/4), (x_2 = (-5 - isqrt(15))/4) :}`