Question

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

Answer 1

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.