What is the discriminant of #5x^2 + 11x + 8 = 0# and what does that mean?

1 Answer
Jul 16, 2015

Answer:

The discriminant is -39. It tells you that there are no real roots to the equation, but there are two complex roots.

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

#5x^2 +11x +8 = 0#

#Δ = b^2 – 4ac = 11^2 -4×5×8 = 121 – 160 = -39#

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

We can see this if we solve the equation.

#5x^2 +11x +8 = 0#

#x = (-b±sqrt(b^2-4ac))/(2a) = (-11±sqrt(11^2 -4×5×8))/(2 ×5) = (-11±sqrt(121-160))/10 = (-11±sqrt(-39))/10 = 1/10(-11 ±isqrt39) = -1/10(11±isqrt39)#

#x = -1/10(11+isqrt39)# and #x = -1/10(11-isqrt39)#

There are no real roots, but there are two complex roots to the equation.