Derivation of Quadratic Formula

A D V E R T I S E M E N T

The roots of a quadratic equation [math]ax^2 + bx + c = 0[/math] 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 [math]ax^2 + bx + c = 0[/math] by [math]a[/math].
Transpose the quantity [math]\dfrac{c}{a}[/math] to the right side of the equation.
Complete the square by adding [math]\dfrac{b^2}{4a^2}[/math] 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 [math]x[/math] by transporting the quantity [math]\dfrac{b}{2a}[/math] to the right side of the equation.
Combine the right side of the equation to get the quadratic formula.

See the derivation below.

[math]ax^2 + bx + c = 0[/math]

[math]x^2 + \dfrac{b}{a}x + \dfrac{c}{a} = 0[/math]

[math]x^2 + \dfrac{b}{a}x = -\dfrac{c}{a}[/math]

[math]x^2 + \dfrac{b}{a}x + \dfrac{b^2}{4a^2} = \dfrac{b^2}{4a^2} - \dfrac{c}{a}[/math]

[math]\left( x + \dfrac{b}{2a} \right)^2 = \dfrac{b^2 - 4ac}{4a^2}[/math]

[math]x + \dfrac{b}{2a} = \dfrac{\pm \sqrt{b^2 - 4ac}}{2a}[/math]

[math]x = -\dfrac{b}{2a} + \dfrac{\pm \sqrt{b^2 - 4ac}}{2a}[/math]

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

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