Summary of the solutions for the quadratic equation $$\Tblue{a}\Tred{x}^2 + \Tblue{b}\Tred{x}+\Tblue{c} = 0$$ for numbers $\Tblue{a}\ne0$, $\Tblue{b}$ and $\Tblue{c}$.
We use the discriminant $\Tviolet{\Delta} = \Tblue{b}^2 - 4 \Tblue{ac}$.
Case Formula of solution Number of solutions $\Tviolet{\Delta}\lt 0$ $\Tviolet{\Delta}= 0$ $\Tviolet{\Delta}\gt 0$ No solution $\displaystyle\frac{-\Tblue{b}}{2\Tblue{a}}$ $\displaystyle\frac{-\Tblue{b}\pm\sqrt{\Tviolet{\Delta}}}{2\Tblue{a}}$ $0$ $1$ $2$ $\Tred{x}^2 + 1 = 0$ $\Tred{x}^2 - 2\Tred{x} + 1 = 0$ $\Tred{x}^2 +2\Tred{x} = 0$ $-1$ $0$ $4$ None $1$ $-2$, $0$