How the Quadratic Formula Works: Derivation, Discriminant & Complex Roots

Any second-degree polynomial equation can be expressed in standard form as: $$ax^2 + bx + c = 0 \quad (a \neq 0)$$ Where $a$, $b$, and $c$ are real constant coefficients, and $x$ represents the unknown variable.

Derived by completing the square on the standard equation, the universal solution formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ The $\pm$ symbol signifies that quadratic equations generally possess two distinct solutions representing where the parabola crosses the horizontal x-axis.

The expression under the radical, $\Delta = b^2 - 4ac$, is known as the discriminant. It reveals the nature of the roots without fully graphing the curve: - **$\Delta > 0$**: Two distinct real roots. - **$\Delta = 0$**: Exactly one repeated real root (the parabola’s vertex touches the x-axis). - **$\Delta < 0$**: Two complex conjugate roots containing the imaginary unit $i = \sqrt{-1}$.