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Solving a quadratic equation (overview)

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 $$\Tviolet{\Delta}\lt 0$$ $$\Tviolet{\Delta}= 0$$ $$\Tviolet{\Delta}\gt 0$$
Formula of solution No solution $$\displaystyle\frac{-\Tblue{b}}{2\Tblue{a}}$$ $$\displaystyle\frac{-\Tblue{b}\pm\sqrt{\Tviolet{\Delta}}}{2\Tblue{a}}$$
Number of solutions $$0$$ $$1$$ $$2$$
Example $$\Tred{x}^2 + 1 = 0$$ $$\Tred{x}^2 - 2\Tred{x} + 1 = 0$$ $$\Tred{x}^2 +2\Tred{x} = 0$$
$$\Delta = $$ $$-1$$ $$0$$ $$4$$
Solutions None $$1$$ $$-2$$, $$0$$