Quadratic formula

The Quadratic Formula is a mathematical formula used to find the x-intercepts for a quadratic function, or parabola. When

$f(x) = ax^2 + bx + c = 0,\quad a \ne 0$

With $x$ being the variable and $a$, $b$ and $c$ being constant, the quadratic equation is:

$\displaystyle x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

This is one possible method to determine the x-intercepts. They can also be found by completing the square, factoring or graphing. For the simpler formula where $a = 0,\quad x = \frac{-c}{b}$.

Discriminant

The number of x-intercepts, or solutions, can be determined by the value of the discriminant:

$b^2 - 4ac$

If the value is positive, there are always two real solutions.

If the value is zero, there is one solution.

If it is negative, there are no real solutions but two complex conjugate solutions.

Derivation by completing the square

Given the quadratic equation $ax^{2} + bx + c = 0$

Divide the equation by a: $x^2+\frac{b}{a}x+\frac{c}{a}=0$

Subtract the constant term c/a from both sides: $x^2+\frac{b}{a}x=-\frac{c}{a}$

Add the square of b/2a to both sides to get a perfect square at the left hand side: $x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=\frac{b^2}{4a^2}-\frac{c}{a}$

Factor the left hand side and simplify the right hand side: $\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$

Take the square root of both sides: $x+\frac{b}{2a}=\frac{\pm\sqrt{b^2-4ac}}{2a}$

Solving for x gives the quadratic formula: $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$