How to Derive the Quadratic Formula


 


Video Transcript

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.



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