basic mathematics image
 basic mathematics image

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


quadratic-formula-proof-image

quadratic-formula-proof-image

Subtract c/a from both sides


quadratic-formula-proof-image

quadratic-formula-proof-image

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:


quadratic-formula-proof-image

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.

Then, square the right side to get (b2)/(4a2)


quadratic-formula-proof-image

Get the same denominator on the right side:


quadratic-formula-proof-image

quadratic-formula-proof-image

Now, take the square root of each side:


quadratic-formula-proof-image


Simplify the left side:


quadratic-formula-proof-image


quadratic-formula-proof-image


Rewrite the right side:


quadratic-formula-proof-image


Subtract b/2a from both sides:


quadratic-formula-proof-image


quadratic-formula-proof-image


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


quadratic-formula-proof-image


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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