SOLUTION: Starting with ax2 + bx + c = 0 and show each step to end up with: x = - b + √b2 - 4ac ___________ ___________ 2a

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Question 113700: Starting with ax2 + bx + c = 0 and show each step to end up with:
x = - b + √b2 - 4ac
_______________________
2a

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
Quadratic Formula Proven
x%5B1%2C2%5D=%28-b+%2B-+sqrt+%28b%5E2+-4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29
Consider equation in a standard form ax%5E2+%2Bbx+%2Bc+=+0++
. What we want to do is complete the square, that is; note that a, b, and c can be positive as well as negative. The right part of the equation should always be zero.
get an equation like this:
(...)^2 = ...
Divide both parts by a:
%28a%2Fa%29x%5E2+%2B+%28b%2Fa%29x+%2B+%28c%2Fa%29+=+0%2Fa++
x%5E2+%2B+%28b%2Fa%29x+%2B+%28c%2Fa%29+=+0++………..we will %28b%2Fa%29x+multiply and divide by 2
x%5E2+%2B+2%28b%2F2a%29x+%2B+%28c%2Fa%29+=+0++………..move c%2Fa to the right
x%5E2+%2B+2%28b%2F2a%29x+=+-+%28c%2Fa%29+++………..
Now, if we want to extract+a+square of a binomial out of here, we can add +%28b%2F2a%29%5E2+ to both sides of the equation:
x%5E2+%2B+2%28b%2F2a%29x+%2B+%28b%2F2a%29%5E2++=+-+%28c%2Fa%29+%2B++%28b%2F2a%29%5E2+++………..
We did it so that the formula to the left would be a complete square of an expression:
%28x+%2B+%28b%2F2a%29%29%5E2++=+-+%28c%2Fa%29+%2B++%28b%2F2a%29%5E2+++………..

On the left side, there is a square of something, on the right, it is a number.
As you know, for all real numbers their squares are non-negative.
So if the number to the right happens to be negative, this means that there+is+no+real+value+of+x that would satisfy this equation.
That's where the discriminat rule comes from.
Since the roots of equations like +y%5E2+=+z+are y%5B1%2C2%5D+=+0+%2B-sqrt%28z%29, we have:
x%5B1%2C2%5D+=+-+%28b%2F2a%29+%2B-+sqrt%28-c%2Fa+%2B+%28b%2F2a%29%5E2%29
This quadratic formula can be simplified as:
open the bracket with %28b%2F2a%29%5E2:
x%5B1%2C2%5D+=+-+%28b%2F2a%29+%2B-+sqrt%28-c%2Fa+%2B+%28b%5E2%2F2a%5E2%29%29
x%5B1%2C2%5D+=+-+%28b%2F2a%29+%2B-+sqrt%28-c%2Fa+%2B+%28b%5E2%2F4a%5E2%29%29
divide and multiply the square root by 2a:
x%5B1%2C2%5D+=+-+%28b%2F2a%29+%2B-2a+sqrt%28-c%2Fa+%2B+%28b%5E2%2F4a%5E2%29%29%2F2a
get 2a inside the square root as 4a%5E2:
x%5B1%2C2%5D+=+-+%28b%2F2a%29+%2B-+sqrt%28-c4a%5E2+%2Fa+%2B+%28b%5E2%2F4a%5E2%29%29%2F2a
use distributive property:

Simplify:
x%5B1%2C2%5D+=+-+%28b%2F2a%29+%2B-+sqrt%28-4ac+%2B+b%5E2%29%2F2a
Or
x%5B1%2C2%5D+=+-+%28b%2F2a%29+%2B-+sqrt%28b%5E2+-+4ac+%29%2F2a
get 2a as the common denominator:
x%5B1%2C2%5D=%28-b+%2B-+sqrt+%28b%5E2+-4%2Aa%2Ac+%29%29+%2F+2a