Any quadratic equation can be solved with the quadratic formula.
$$x=\frac{-\textcolor{#5cb85c}{b} \pm \sqrt{\textcolor{#5cb85c}{b}^2-4\textcolor{#2d6da3}{a}\textcolor{#d9534f}{c}}}{2\textcolor{#2d6da3}{a}}$$
But where did the quadratic formula come from?
Let me show you how you could derive the quadratic formula for yourself.
Let's start with:
$$\textcolor{#2d6da3}{a}x^2+\textcolor{#5cb85c}{b}x+\textcolor{#d9534f}{c}=0$$
Let's divide both sides by a:
$$x^2+\frac{\textcolor{#5cb85c}{b}}{\textcolor{#2d6da3}{a}}x+\frac{\textcolor{#d9534f}{c}}{\textcolor{#2d6da3}{a}}=0$$
Let's now subtract c/a from both sides:
$$x^2+\frac{\textcolor{#5cb85c}{b}}{\textcolor{#2d6da3}{a}}x=-\frac{\textcolor{#d9534f}{c}}{\textcolor{#2d6da3}{a}}$$
Now to complete the square let's add (b/(2a))^2 to both sides:
$$x^2+\frac{\textcolor{#5cb85c}{b}}{\textcolor{#2d6da3}{a}}x+(\frac{\textcolor{#5cb85c}{b}}{2\textcolor{#2d6da3}{a}})^2=-\frac{\textcolor{#d9534f}{c}}{\textcolor{#2d6da3}{a}}+(\frac{\textcolor{#5cb85c}{b}}{2\textcolor{#2d6da3}{a}})^2$$
We see that the left side factors into:
$$(x+\frac{\textcolor{#5cb85c}{b}}{2\textcolor{#2d6da3}{a}})^2=-\frac{\textcolor{#d9534f}{c}}{\textcolor{#2d6da3}{a}}+(\frac{\textcolor{#5cb85c}{b}}{2\textcolor{#2d6da3}{a}})^2$$
Let's simplify the right hand side:
$$(x+\frac{\textcolor{#5cb85c}{b}}{2\textcolor{#2d6da3}{a}})^2=\frac{\textcolor{#5cb85c}{b}^2-4\textcolor{#2d6da3}{a}\textcolor{#d9534f}{c}}{4\textcolor{#2d6da3}{a}^2}$$
We can take the square root of both sides:
$$x+\frac{\textcolor{#5cb85c}{b}}{2\textcolor{#2d6da3}{a}}=\pm\sqrt{\frac{\textcolor{#5cb85c}{b}^2-4\textcolor{#2d6da3}{a}\textcolor{#d9534f}{c}}{4\textcolor{#2d6da3}{a}^2}}$$
$$x+\frac{\textcolor{#5cb85c}{b}}{2\textcolor{#2d6da3}{a}}=\pm\frac{\sqrt{\textcolor{#5cb85c}{b}^2-4\textcolor{#2d6da3}{a}\textcolor{#d9534f}{c}}}{2\textcolor{#2d6da3}{a}}$$
To get x by itself we can then subtract b/(2a) from both sides:
$$x=\pm\frac{\sqrt{\textcolor{#5cb85c}{b}^2-4\textcolor{#2d6da3}{a}\textcolor{#d9534f}{c}}}{2\textcolor{#2d6da3}{a}}-\frac{\textcolor{#5cb85c}{b}}{2\textcolor{#2d6da3}{a}}$$
After we simplify the right hand side:
$$x=\frac{-\textcolor{#5cb85c}{b} \pm \sqrt{\textcolor{#5cb85c}{b}^2-4\textcolor{#2d6da3}{a}\textcolor{#d9534f}{c}}}{2\textcolor{#2d6da3}{a}}$$
we have derived the quadratic formula.
Solve x^2+4x+3=0: x^2+4x+3=0
Solve 3x^2+4x=2x^2-3: 3x^2+4x=2x^2-3
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