What is the discriminant of #x^2+x+1=0# and what does that mean?

1 Answer
Jul 16, 2015

Answer:

The discriminant is -3. It tells you that there are no real roots, but there are two complex roots to the equation.

Explanation:

If you have a quadratic equation of the form

#ax^2+bx+c=0#

The solution is

#x = (-b±sqrt(b^2-4ac))/(2a)#

The discriminant #Δ# is #b^2 -4ac#.

The discriminant "discriminates" the nature of the roots.

There are three possibilities.

  • If #Δ > 0#, there are two separate real roots.
  • If #Δ = 0#, there are two identical real roots.
  • If #Δ <0#, there are no real roots, but there are two complex roots.

Your equation is

#x^2 +x +1 = 0#

#Δ = b^2 – 4ac = 1^2 - 4×1×1 = 1 – 4 = -3#

This tells you that there are no real roots, but there are two complex roots.

We can see this if we solve the equation.

#x^2 +x +1 = 0#

#x = (-b±sqrt(b^2-4ac))/(2a) = (-1±sqrt(1^2 - 4×1×1))/(2×1) = (-1±sqrt(1-4))/2 = (-1 ±sqrt(-3))/2 = 1/2(-1±isqrt3) =-1/2(1±isqrt3)#

#x =—1/2(1+ isqrt3)# and #x = -1/2(1- isqrt3)#