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

1 Answer
Jul 27, 2015

Answer:

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) :}#