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

1 Answer
Stefan V.
Jul 27, 2015

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) :}

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