What is the discriminant of #2x^2 - 3x + 4 = 0# and what does that mean?

1 Answer
Jul 16, 2015

Answer:

The discriminant is -23. It tells you that there are no real roots to the equation, but there are two separate 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

#2x^2 – 3x +4 = 0#

#Δ = b^2 – 4ac = (-3)^2 -4×2×4 = 9 – 32 = -23#

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

We can see this if we solve the equation.

#2x^2–3x+4 = 0#

#x = (-b±sqrt(b^2-4ac))/(2a) = (-(-3)±sqrt((-3)^2 -4×2×4))/(2×2) = (3±sqrt(9-32))/4 = (3±sqrt(-23))/4 = 1/4(3±isqrt23)#

#x = 1/4(3+isqrt23)# and #x = 1/4(3-isqrt23)#

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