# Derivation of Quadratic Formula

The roots of a quadratic equation *ax*^{2} + *bx* + *c* = 0 is given by the quadratic formula

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

The derivation of this formula can be outlined as follows:

- Divide both sides of the equation
*ax*^{2}+*bx*+*c*= 0 by*a*. - Transpose the quantity
*c*/*a*to the right side of the equation. - Complete the square by adding
*b*^{2}/ 4*a*^{2}to both sides of the equation. - Factor the left side and combine the right side.
- Extract the square-root of both sides of the equation.
- Solve for
*x*by transporting the quantity*b*/ 2*a*to the right side of the equation. - Combine the right side of the equation to get the quadratic formula.

See the derivation below.

**Derivation of Quadratic Formula**

$ax^2 + bx + c = 0$

$x^2 + \dfrac{b}{a}x + \dfrac{c}{a} = 0$

$x^2 + \dfrac{b}{a}x = -\dfrac{c}{a}$

$x^2 + \dfrac{b}{a}x + \dfrac{b^2}{4a^2} = \dfrac{b^2}{4a^2} - \dfrac{c}{a}$

$\left( x + \dfrac{b}{2a} \right)^2 = \dfrac{b^2 - 4ac}{4a^2}$

$x + \dfrac{b}{2a} = \dfrac{\pm \sqrt{b^2 - 4ac}}{2a}$

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

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

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