SOLUTION: In "completing the square" to derive the quadratic equation, there is a step in which the square root is taken of both sides of the equation. This yields x + b/(2a) on the left. Me

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Question 881195: In "completing the square" to derive the quadratic equation, there is a step in which the square root is taken of both sides of the equation. This yields x + b/(2a) on the left. Meanwhile on the right, the square root of the numerator (b^2 - 4ac) is taken separately from the square root of the denominator (4a^2), so that the denominator can be expressed as 2a. The square root in the numerator is left in the finished quadratic formula and given a +/- symbol. I was wondering if anyone could help me understand why the square root in the denominator is 2a instead of +/-(2a). Why +/- in the numerator but NOT the denominator?
Found 2 solutions by josgarithmetic, KMST:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
See and study this lesson which will explain Completing the Square very clearly:

What is Completing The Square? With Visual Explanation

You could also use another lesson after that one:
Completing the Square to Solve General Quadratic Equation

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
You have
%28x%2Bb%2F2a%29%5E2=b%5E2%2F4a%5E2-c%2Fa-->%28x%2Bb%2F2a%29%5E2=b%5E2%2F4a%5E2-4ac%2F4a%5E2-->%28x%2Bb%2F2a%29%5E2=%28b%5E2+-+4ac%29%2F4a%5E2
%28x%2Bb%2F2a%29%5E2 is the square of x%2Bb%2F2a and it is also the square of -%28x%2Bb%2F2a%29 .
One of those expressions may be positive and the other negative,
but we pick one, and we prefer to pick the one that does not start with a minus sign, because x%2Bb%2F2a is simpler to write.
The other side of the equal sign could also be the square of 2 different expressions. One of them will be equal to x%2Bb%2F2a .
%28b%5E2+-+4ac%29%2F4a%5E2 is the square of sqrt%28b%5E2+-+4ac%29%2F2a=-sqrt%28b%5E2+-+4ac%29%2F%28-2a%29 and
it is also the square of -sqrt%28b%5E2+-+4ac%29%2F2a=sqrt%28b%5E2+-+4ac%29%2F%28-2a%29 .
You can write 4 different ways the expressions that squared equal %28b%5E2+-+4ac%29%2F4a%5E2 ,
but they represent only 2 different possibilities.
It's like flipping a coin: it could be heads=not-tails or tails=not-heads.
Of course, given a choice, we prefer not to have denominators that start with a minus sign.
So we say that
either
x%2Bb%2F2a=sqrt%28b%5E2-4ac%29%2F2a-->x=-b%2F2a%2Bsqrt%28b%5E2-4ac%29%2F2a-->x=-b%2Bsqrt%28b%5E2-4ac%29%29%2F2a
or
x%2Bb%2F2a=-sqrt%28b%5E2-4ac%29%2F2a-->x=-b%2F2a-sqrt%28b%5E2-4ac%29%2F2a-->x=-b-sqrt%28b%5E2-4ac%29%29%2F2a