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


Proof of the quadratic formula
Proof of the quadratic formula


Subtract c/a from both sides


Proof of the quadratic formula
Proof of the quadratic formula


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:


Proof of the quadratic formula


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.

Proof of the quadratic formula
Get the same denominator on the right side:

Proof of the quadratic formula
Proof of the quadratic formula


Now, take the square root of each side:


Proof of the quadratic formula


Simplify the left side:


Proof of the quadratic formula
Proof of the quadratic formula


Rewrite the right side:


Proof of the quadratic formula


Subtract b/2a from both sides:


Proof of the quadratic formula
Proof of the quadratic formula


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


Proof of 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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