# Proof of the quadratic formula

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

ax2 + bx + c = 0

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

Subtract c/a from both sides

Complete the square:

The coefficient of the second term is b/a

Divide this coefficient by 2 and square the result to get (b/2a)2

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(b/2a)

Add these two and raise everything to the second.

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 b/2a from both sides:

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

The + - 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.

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