Question

What is the discriminant of `x^2+x+1=0` and what does that mean?

Answer 1

The discriminant is -3. It tells you that there are no real roots, but there are two complex roots to the equation.

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

`x^2 +x +1 = 0`

`Δ = b^2 – 4ac = 1^2 - 4×1×1 = 1 – 4 = -3`

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

We can see this if we solve the equation.

`x^2 +x +1 = 0`

`x = (-b±sqrt(b^2-4ac))/(2a) = (-1±sqrt(1^2 - 4×1×1))/(2×1) = (-1±sqrt(1-4))/2 = (-1 ±sqrt(-3))/2 = 1/2(-1±isqrt3) =-1/2(1±isqrt3)`

`x =—1/2(1+ isqrt3)` and `x = -1/2(1- isqrt3)`