What is the discriminant of #3x^2 - 5x + 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 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

#3x^2 – 5x +4 = 0#

#Δ = b^2 – 4ac = (-5)^2 -4×3×4 = 25 – 48 = -23#

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

We can see this if we solve the equation.

#3x^2 – 5x +4 = 0#

#x = (-b±sqrt(b^2-4ac))/(2a) = (-(-5)±sqrt((-5)^2 -4×3×4))/(2 ×3) = (5±sqrt(25-48))/6 = (5±sqrt(-23))/6 = 1/6(5 ±isqrt23)#

#x = 1/6(5+isqrt23)# and #x = 1/6(5-isqrt23)#

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