Algebra - Quadratics

Quadratic Equations and the Discriminant

A quadratic equation has the standard form $ax^2 + bx + c = 0$, where $a \neq 0$. When factorising by inspection or the guide number method is awkward, we use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The term under the square root, $\Delta = b^2 - 4ac$, is the discriminant. It tells us the nature of the roots without having to solve the equation completely.

If $b^2 - 4ac > 0$, the equation has two distinct real roots, meaning the graph crosses the x-axis twice. If $b^2 - 4ac = 0$, the equation has one repeated real root, meaning the curve touches the x-axis at its turning point. If $b^2 - 4ac < 0$, there are no real roots, so the parabola never meets the x-axis.

For example, if you are asked to find the values of $m \in \mathbb{Z}$ for which $3x^2 - mx + 3 = 0$ has exactly one solution, set the discriminant equal to zero. Here $a = 3$, $b = -m$, and $c = 3$. Evaluating $(-m)^2 - 4(3)(3) = 0$ gives $m^2 - 36 = 0$, which yields $m = 6$ or $m = -6$. If roots $\alpha$ and $\beta$ are given, you can rebuild the quadratic using the structure $x^2 - (\alpha + \beta)x + \alpha\beta = 0$.