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.






Page copy protected against web site content infringement by Copyscape





worksheets-banner










Copyright © 2008-2015. Basic-mathematics.com. All right reserved

Are you a fan of this site? Support us     Our awards!     Our partners     About me     Disclaimer     Build your website!     Advertise on my site
    Try our free toolbar     Like us on Facebook     Take our survey     Illustrated fractions     Educational math software     Math jobs     Best Teacher Sites Now     Teachers/Students tools     Search math jobs     Algebra ebook     Fraction ebook     Geometric formulas ebook