# How do you use the discriminant to determine the numbers of solutions of the quadratic equation z^2 + z + 1 = 0 and whether the solutions are real or complex?

Jul 8, 2018

no real solutions

#### Explanation:

Given: ${z}^{2} + z + 1 = 0$

When the equation is in the form: $A {z}^{2} + B z + C = 0$ the discriminant is ${B}^{2} - 4 A C$

When ${B}^{2} - 4 A C = 0$ there is 1 real solution

When ${B}^{2} - 4 A C > 0$ and not a perfect square: 2 real solutions

When ${B}^{2} - 4 A C < 0$ there are no solutions, the answers are imaginary.

For the given equation:

${B}^{2} - 4 A C = 1 - 4 \left(1\right) \left(1\right) = 1 - 4 = - 3 \implies$no real solutions