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# Proof of the Quadratic Formula

Posted Feb23-10 at 08:08 PM by jhae2.718
Updated May19-11 at 06:07 PM by jhae2.718

I was reading the Library article on quadratic equations--I'm not sure why--and noticed that a proof of the quadratic formula is not offered. Here is a proof that depresses the quadratic into an equation of the form ax2=b, where a and b are some arbitrary constants.

For the general quadratic equation ax2+bx+c=0:

Let $$x=y-\frac{b}{2a}$$
$$a \left(y-\frac{b}{2a} \right)^2 +b\left(y-\frac{b}{2a}\right)+c=0$$
Expand $$\left(y-\frac{b}{2a}\right)^2$$ and multiply by a:
$$ay^2-by+\frac{b^2}{4a}+by-\frac{b^2}{2a}+c=0$$
Eliminate the by terms
\begin{align*} ay^2+\frac{b^2}{4a}-\frac{b^2}{2a}+c &= 0 \\ ay^2 &= \frac{b^2}{2a}-\frac{b^2}{4a}-c \\ y^2 &= \frac{b^2}{2a^2}-\frac{b^2}{4a^2}-\frac{c}{a} \\ y^2 &= \frac{b^2-4ac}{4a^2} \\ y &= \frac{\pm\sqrt{b^2-4ac}}{2a} \end{align*}

Since x=y-b/(2a),
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

Q.E.D.

Personally, I find this method more appealing than completing the square.
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