Quadratic Formula's proof

Have you ever wondered why the quadratic formula is like that?

In this page, you will learn how to prove quadratic formula and also why it is in that form.

\[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \]

The quadratic formula

Warning

To prove the quadratic equation, we are going to use completion of the square method. Since this method is probably the easiest and most intitive way of proving, we recommend learning this method before continuing further.

Step 1:

Factor out a.

\[ a(x^2+\frac{b}{a})+c=0 \]

Step 2:

Apply the completion of square. Insert (b/2a)^2 to do this.

\[ a(x^2+\frac{b}{a}+(\frac{b}{2a})^2)-(\frac{b}{2a})^2+c=0 \]

Step 3:

Simplify.

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

Step 4:

Factor.

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

Step 5:

Divide both sides by a.

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

Step 6:

Simplify the right side.

\[ (x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2} \]

Step 7:

Take the square root of both sides.

\[ x+\frac{b}{2a}=\pm\sqrt{\frac{b^2-4c}{4a^2}}=\pm\frac{\sqrt{b^2-4ac}}{2a} \]

Info

This is where it leaves us with discriminant. ※The numerator.

Step 8:

We are almost done!!

The last step is to move the b/2a term.

\[ x=\frac{\pm\sqrt{b^2-4ac}}{2a}-\frac{b}{2a}=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \]

And this is the complete proof to the quadratic formula!!

If you have time, try using pencil and paper to figure this out again on your own.