The following is a proof of the quadratic formula. It will show you how the quadratic formula that is widely used was developed.

The proof is done using the standard form of a quadratic equation and solving the standard form by completing the square

Start with the the standard form of a quadratic equation:

The proof is done using the standard form of a quadratic equation and solving the standard form by completing the square

Start with the the standard form of a quadratic equation:

Divide both sides of the equation by a so you can complete the square

Subtract $\frac{c}{a}$ from both sides

Complete the square:

The coefficient of the second term is $\frac{b}{a}$

Divide this coefficient by 2 and square the result to get $\left (\frac{b}{2a} \right )^2$

Add $\left (\frac{b}{2a} \right )^2$ to both sides:

The coefficient of the second term is $\frac{b}{a}$

Divide this coefficient by 2 and square the result to get $\left (\frac{b}{2a} \right )^2$

Add $\left (\frac{b}{2a} \right )^2$ to both sides:

Since the left side of the equation right above is a perfect square, you can factor the left side by using the coefficient of the first term ($x$) and the base of the last term $\frac{b}{2a}$

Add these two and raise everything to the second.

Then, square the right side to get $\frac{b^2}{4a^2}$

Add these two and raise everything to the second.

Then, square the right side to get $\frac{b^2}{4a^2}$

Get the same denominator on the right side:

Now, take the square root of each side:

Simplify the left side:

Rewrite the right side:

Subtract $\frac{b}{2a}$ from both sides:

Adding the numerator and keeping the same denominator, we get the quadratic formula:

The $\pm$ between the $b$ and the square root sign means plus or negative. In other words, most of the time, you will get two answers when using the quadratic formula.