Question

What is the discriminant of `2x^2 - 3x + 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 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.