Quadratic Formula Derivation

The standard form of a quadratic equation is:

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

Here, $a, b, c \in ℝ$ , and $a \neq 0$

We begin by first dividing both sides by the coefficient $a$

⟹ x^{2} + \frac{b}{a} x + \frac{c}{a} = 0

We can rearrange the equation:

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

We can then use the method of completing the square. (Maths is Fun has a really good explanation for this technique)

x^{2} + \frac{b}{a} x + (\frac{b}{2 a})^{2} = \frac{- c}{a} + (\frac{b}{2 a})^{2}

On our LHS, we can clearly recognize that it is the expanded form of $(x + d)^{2}$ i.e $x^{2} + 2 x \cdot d + d^{2}$

⟹ (x + \frac{b}{2 a})^{2} = \frac{- c}{a} + \frac{b^{2}}{4 a^{2}} = \frac{- 4 a c + b^{2}}{4 a^{2}}

Taking the square root of both sides

\begin{matrix} x + \frac{b}{2 a} & = \frac{\sqrt{- 4 a c + b^{2}}}{2 a} \\ x & = \frac{\pm \sqrt{- 4 a c + b^{2}} - b}{2 a} \\ = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{matrix}

This gives you the world famous quadratic formula:

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

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