The standard form of a quadratic equation is:

$a{x}^{2}+bx+c=0$

Here, $a,b,c\in ℝ$, and $a\ne 0$

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)

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

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

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

Taking the square root of both sides

$\begin{array}{cc}\hfill x+\frac{b}{2a}& =\frac{\sqrt{-4ac+{b}^{2}}}{2a}\hfill \\ \hfill x& =\frac{±\sqrt{-4ac+{b}^{2}}-b}{2a}\hfill \\ \hfill & =\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}\hfill \end{array}$

This gives you the world famous quadratic formula:

$x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$
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