# 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

Start with the the standard form of a quadratic equation:

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

Add (b/2a)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(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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