Proof of the quadratic formula

The following is a proof of the quadratic formula, probably the most important formula in high school. 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 that 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.

This stuff is still complicated! Is there an easier way to come up with a proof of the quadratic formula ? Why do I need to bother myself with this?

Unfortunately, there is no easy shortcut to take that will allow you to derive the quadratic formula. The University of Georgia has dedicated a page to show the different ways to derive the quadratic formula. Check it out and see if you understand these proofs as well.

The site also explains if it is important for teachers to teach the proof of the quadratic formula to students. As a teacher myself, I will definitely say it is a must once students understand clearly how to solve a quadratic equation by completing the square. Moreover, it is requirement that students know how to prove or derive the quadratic formula according to the common core which is the ultimate authority on this matter.

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