What is Root, discriminant? — Mathematics

Definition 10.6 High School Mathematics · Chapter 10 — Quadratic Functions and Equations

A root of the trinomial $ax^2+bx+c$ is a solution of the equation $ax^2 + bx + c = 0$ . The discriminant of the trinomial is the number

\Delta = b^2 - 4ac .

Examples

Example 10.8

Solve $2x^2 - 8x + 3 = 0$ . Here $\Delta = (-8)^2 - 4 \times 2 \times 3 = 64 - 24 = 40 > 0$ , so there are two roots:

x_{1,2} = \frac{8 \pm \sqrt{40}}{4} = \frac{8 \pm 2\sqrt{10}}{4} = 2 \pm \frac{\sqrt{10}}{2}.

For $x^2 + x + 1 = 0$ : $\Delta = 1 - 4 = -3 < 0$ , no real solution.

Example 10.13

Solve $-x^2 + 3x + 4 \geq 0$ . Here $\Delta = 9 + 16 = 25$ , and the roots are $\frac{-3 \pm 5}{-2}$ , that is $x_1 = -1$ and $x_2 = 4$ . Since $a = -1 < 0$ , the trinomial is negative outside the roots and positive between them. With the wide inequality, the solution set is $\intcc{-1}{4}$ .

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