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Discriminant of a quadratic equation

The discriminant $$\Tviolet{\Delta}$$ of a quadratic equation $$$a\Tred{x}^2 + b\Tred{x}+c = 0$$$ is the number $$$\Tviolet{\Delta} = b^2 - 4 ac.$$$ The computation of the discriminant is essential for solving a quadratic equation.

The discriminant of the equation $$\Tred{x}^2 - 3\Tred{x} + 2 = 0 $$ is $$$ \Tviolet{\Delta} = 3^2 - 4\times 1\times 2 = 9-8 = 1.$$$

The discriminant can be positive, negative or zero. The sign of the discriminant tells us whether an equation can be solved.

Equation Discriminant Sign Number of solutions
$$x^2 - 3x + 1 = 0$$ $$9 - 4 = 5$$ $$\gt 0$$ $$2$$
$$x^2 - 2x + 1 = 0$$ $$4 - 4 = 0$$ $$= 0$$ $$1$$
$$x^2 - x + 1 = 0$$ $$1 - 4 = -3$$ $$\lt 0$$ $$0$$