Derivation of Quadratic Formula

The roots of a quadratic equation ax2 + 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:

  1. Divide both sides of the equation ax2 + bx + c = 0 by a.
  2. Transpose the quantity c/a to the right side of the equation.
  3. Complete the square by adding b2 / 4a2 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 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
$ax^2 + bx + c = 0$

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

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

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

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

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

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

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

 

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