Question 19451: I encounted in a math book recently the following equation:
x^2+x-x^2
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sqrt x^2+x+x
However, only the first two terms in the denominator are under the radical.
According to this book the above equation becomes:
x
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x(sqrt 1+1/x+1)
Again, however, the third term in the denominator is not under the radical.
Concerning the numerator, I understand why the previous equation's numerator is x. But I don't really know why and how the denominator in the previous equation is derived. The author says "Factor the x out of the denominator" but I am confused about what specific steps are needed to get the result in this denominator.
A math friend of mine suggested that I factor x^2 out of sqrt x^2+x under the radical. I assume that I could make the sqrt x^2 under the denominator first into sqrt (x^2)(1)to get x(sqrt 1... concerning this first term, but this still leaves me figuring out how the second term becomes 1/x. I imagine this is another way of saying that I do not know how x (sqrt 1+1/x...can become, in reverse, sqrt X^2+x... My math friend said that I should change the left factor of the previous denominator to sqrt x^2 and then multiply the root of x^2 times the equation in the parenthesis instead of x times the equation in the parenthesis. But if I did that, this would result in the third term,1, being x^2 rather than only 1.
So, as one can see, I am confused about this matter.
I would appreciate if someone could help clear up my confusion by explaining in a step-by-step fashion how to factor x out of the denominator in the intial equation to get the supposedly correct result in the denominator in the subsequent equation.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! YOUR SINCERE DESIRE TO TRY BY YOUR SELF FIRST AND ASK FOR CLARRIFICATIONS IS REALLY IMPRESSIVE.YOU WILL CERTAINLY SUCCEED WITH THIS APPROACH...............MY COMMENTS ARE GIVEN IN BOLD
x^2+x-x^2
_________________YOU BETER TYPE THIS USING BRACKETS AS FOLLOWS..IN VIEW OF sqrt x^2+x+x ....... WHAT YOU SAID BELOW ..sqrt(x^2+x)+x
However, only the first two terms in the denominator are under the radical.
According to this book the above equation becomes:
x GOOD
___________________
x(sqrt 1+1/x+1) TYPE AS x(sqrt(1+1/x)+1)
Again, however, the third term in the denominator is not under the radical.
Concerning the numerator, I understand why the previous equation's numerator is x. VERY GOOD
But I don't really know why and how the denominator in the previous equation is derived. The author says "Factor the x out of the denominator" but I am confused about what specific steps are needed to get the result in this denominator....OK LET ME EXPLAIN....WE FIND x IN NR,WE HAVE x BY SIDE OF SQRT TERM IN DR..SO IF WE CAN TAKE OUT x FROM SQRT TERM ALSO ,WE CAN CANCEL x THROUGH OUT...BUT TO TAKE x FROM SQUARE ROOT ,WE HAVE TO TAKE x^2 AS SQRT OF x^2=x..SO WE WRITE sqrt(x^2+x) IN DR AS sqrt(x^2(x^2/x^2 +x/x^2))
=sqrt(x^2(1+1/x))=sqrtx^2*sqrt(1+1/x)=xsqrt(1+1/x) ..HOPE IT IS CLEAR
A math friend of mine suggested that I factor x^2 out of sqrt x^2+x under the radical.
CORRECT
I assume that I could make the sqrt x^2 under the denominator first into sqrt (x^2)(1)
CORRECT UP TO THIS
to get x(sqrt 1... concerning this first term, but this still leaves me figuring out how the second term becomes 1/x.NOW IT IS CLEAR I SUPPOSE
SO FINALLY WE GOT NOW IN THE DR sqrt(x^2+x)+x = xsqrt(1+1/x)+x=x(sqrt(1+1/x))+1)
=x(1+sqrt(1+1/x))...hence the given expression is
x^2+x-x^2
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sqrt x^2+x+x
=x/x(1+sqrt(1+1/x))
= 1/(1+sqrt(1+1/x))
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