Quadratic Formula

The quadratic formula gives the solutions to any quadratic equation $ax^2 + bx + c = 0$ where $a \neq 0$:

The discriminant

The expression under the square root, $\Delta = b^2 - 4ac$, is called the discriminant:

• $\Delta > 0$: two distinct real roots
• $\Delta = 0$: one repeated real root
• $\Delta < 0$: two complex conjugate roots

Derivation by completing the square

Starting from $ax^2 + bx + c = 0$:

1. Divide by $a$: $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$
2. Move constant: $x^2 + \frac{b}{a}x = -\frac{c}{a}$
3. Complete the square: $\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$
4. Take square root: $x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2a}$
5. Solve for $x$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Vieta's formulas

For roots $x_1, x_2$: