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

Jul 16, 2015

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

$a {x}^{2} + b x + c = 0$

The solution is

x = (-b±sqrt(b^2-4ac))/(2a)

The discriminant Δ is ${b}^{2} - 4 a c$.

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.

${x}^{2} + x + 1 = 0$

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 $x = - \frac{1}{2} \left(1 - i \sqrt{3}\right)$