         The Dragon's Demesne Proof - The Quadratic Formula             Home | Education | Recreation     Proof of the Quadratic Formula               A quadratic equation is an equation of the form Ax^2 + Bx + C = 0 (where the symbol ^ is used to indicate the following number is an exponent) A, B, and C are constant numbers, like 1, 3, 12, 200, 0, 7, etc, and the 'x' represents the solution to this equation. There are many ways to solve a quadratic equation, but the use of the quadratic formula is the easiest, because once the formula is derived, you need only substitute the known values of A, B, and C into the formula, which will then solve explicit values for 'x'. We want to rearrange the equation so that we have x = (something in terms of A,B,C) Proof: We start with the general form of the equation given above. We multiply both sides of the equation by 4A. This gives: 4(Ax)^2 + 4ABx + 4AC = 0 Next we add B^2 to both sides of the equation. 4(Ax)^2 + 4ABx + 4AC + B^2 = B^2 Next we subtract 4AC from both sides of the equation. 4(Ax)^2 + 4ABx + B^2 = B^2 - 4AC Next comes the key part of the proof. The left hand side of this equation is equal to (2Ax + B)(2Ax + B) or (2Ax + B)^2. If you don't see this, you should multiply (2Ax + B) by itself to see that it works. After simplifying the left side of the equation, we have: (2Ax + B)^2 = B^2 - 4AC The next step is to take the square root of both sides of the equation. 2Ax + B = +/- sqrt(B^2 - 4AC) I am using sqrt to represent taking the square root of B^2 - 4AC. The +/- indicates that the right hand side can be either positive or negative. This is a property of taking the square root of a number. For example, if you take 3 times 3, you get 9. But, if you take (-3) times (-3), you also get 9. That is the idea used here, just with symbols. The equation is fairly easy to solve after this using simple algebra. We subtract B from both sides of the equation, like so: 2Ax = -B +/- sqrt(B^2 - 4AC) And then we divide both sides of the equation by 2A/ x = [ -B +/- sqrt(B^2 - 4AC) ] / (2A) Notice that the left hand side of the equation has just 'x', and that the right hand side of the equation has no 'x'. This means that if we know what A, B and C are, that we can find what 'x' is every single time.               Enter supporting content here Please feel free to use any information you find on this site in any way you see fit.  Nothing here is copyrighted unless specifically noted.     