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Derivation of Quadratic Formula

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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
$ax^2 + bx + c = 0$

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

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

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

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

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

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

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

 

Tags: 

Algebra

quadratic formula

Derivation of Formula

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