What is the discriminant of #3x^2 - 5x + 4 = 0# and what does that mean?

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Answer 1 · 2015-07-16T18:52:43

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

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.

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.