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

The derivation of this formula can be outlined as follows:

1. Divide both sides of the equation ax^2 + bx + c = 0 by a.
2. Transpose the quantity \dfrac{c}{a} to the right side of the equation.
3. Complete the square by adding \dfrac{b^2}{4a^2} to both sides of the equation.
4. Factor the left side and combine the right side.
5. Extract the square-root of both sides of the equation.
6. Solve for x by transporting the quantity \dfrac{b}{2a} to the right side of the equation.
7. 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} + \dfrac{\pm \sqrt{b^2 - 4ac}}{2a}