Mastering Quadratic Equations: Formulas, Roots & Graphs

In algebra, a quadratic equation is any equation that can be rearranged in standard form as:

Where $x$ represents an unknown variable, and $a$, $b$, and $c$ represent known coefficients, with $a \neq 0$. If $a = 0$, the equation becomes linear.

The Quadratic Formula

To find the roots of any standard quadratic equation, we use the quadratic formula:

The Discriminant ($D$)

The term inside the square root, $D = b^2 - 4ac$, is known as the discriminant. It determines the nature of the roots:

1. If $D > 0$, the equation has two distinct real roots.
2. If $D = 0$, the equation has one repeated real root (also known as coincident roots).
3. If $D < 0$, the roots are complex conjugate pairs, given by:

Sum and Product of Roots

If the roots are denoted as $\alpha$ and $\beta$, the relations between the roots and coefficients are:

• Sum of roots: $\alpha + \beta = -\frac{b}{a}$
• Product of roots: $\alpha \cdot \beta = \frac{c}{a}$