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

1 Answer
Jul 16, 2015

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.