Question 720317
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Here is my problem.

[SQRT(x + 7)] - 2[SQRT(x)] =-2   OR   &#8730;(x + 7) - 2&#8730;(x) = -2 
-------
Now, the steps I took to solve this problem are to first square both sides:
[&#8730;(x + 7) - 2&#8730;(x)] * [&#8730;(x + 7) - 2&#8730;(x)] = 4
so then I FOIL the left side, resulting in:
(x+7) - [&#8730;(x + 7) * - 2&#8730;(x)] - [ - 2&#8730;(x) * &#8730;(x + 7)] + 4x
So then I thought to subtract (x+7) and (4x) to both sides
 - [&#8730;(x + 7) * - 2&#8730;(x)] - [ - 2&#8730;(x) * &#8730;(x + 7)] = 4 - 4x - x - 7 (I think I'm supposed to switch the sign, because I've
subtracted it and moved it to the opposite side, right?)
I think I'm correct up to this point, but now I have to square both sides again.
I think this left hand side could be re-written as:
 -2[&#8730;(x + 7) * - 2&#8730;(x)]
Is this right? I'm subtracting it from itself, a negative, which could simply multiplied by -2. Anyway, now I need to
square this again, so I assume the -2 becomes a 4 and I FOIL them separately?
FOILING the left side will get:
 [&#8730;(x + 7) * - 2&#8730;(x)] * [&#8730;(x + 7) * - 2&#8730;(x)]
Which, when FOILed, looks like
(x+7) - [&#8730;(x+7) * -2&#8730;(x)] - [ - 2&#8730;(x) * &#8730;(x + 7)] + 4x
It looks exactly the same as before!! I'm just really confused by this problem, and I have a couple more like it, so I
want to know if figuring this one out could help me solve the other ones.

I'm confused about FOILing the different sides, whether or not I can combine two square roots, and quite frankly, a lot
of other things.
One of the options on the test is 9, and I think this is the answer, because I've inserted it into the original
equation and it works, but I'm just confused about how to actually get 9 out of this..

Sorry for the long question. I hope it's not hard to understand. I'm just hoping someone can walk me through all
the steps of solving a problem like this so I can do it easily in the future.
*******************************<font color = blue><font  size = 2><font face = tahoma><b>
Here is my problem.
[SQRT(x + 7)] - 2[SQRT(x)] =-2      OR       &#8730;(x + 7) - 2&#8730;(x) = -2 
-------
Now, the steps I took to solve this problem are to first square both sides:
[&#8730;(x + 7) - 2&#8730;(x)] * [&#8730;(x + 7) - 2&#8730;(x)] = 4
so then I FOIL the left side, resulting in:
(x+7) - [&#8730;(x + 7) * - 2&#8730;(x)] - [ - 2&#8730;(x) * &#8730;(x + 7)] + 4x <font color = red><=== This is where you went WRONG! When FOILed, this's
actually: {{{(x + 7) + sqrt(x + 7) * - 2sqrt(x) - 2sqrt(x) * sqrt(x + 7) + 4x)}}}, which results in:
                 {{{x + 7 - 2sqrt(x)sqrt(x + 7) - 2sqrt(x)sqrt(x + 7) + 4x = 4}}}. See?</font>          
                                     {{{x + 7 - 4sqrt(x)sqrt(x + 7) + 4x = 4}}}                                                                                        
                                                    {{{- 4sqrt(x)sqrt(x + 7) = 4 - (x + 7) - 4x}}}  So then I thought to subtract (x+7) and (4x) to both sides
                                                    {{{- 4sqrt(x)sqrt(x + 7) = 4 - x - 7 - 4x)}}}  - [&#8730;(x + 7) * - 2&#8730;(x)] - [ - 2&#8730;(x) * &#8730;(x + 7)] = 4 - 4x - x - 7
                                                                                                     (I think I'm supposed to switch the sign, because I've 
                                                                                                     subtracted it and moved it to the opposite side, right?)
                                                  {{{- 4sqrt(x^2 + 7x) = - 3 - 5x}}} <font color = red><=== This is exactly how you need to proceed!</font>
                                           {{{(- 4sqrt(x^2 + 7x))^2 = (- 3 - 5x)^2}}} ---- Squaring both sides
                                                 {{{16(x^2 + 7x) = 9 + 30x + 25x^2}}}
                                                  {{{16x^2 + 112x = 9 + 30x + 25x^2}}}
                       {{{16x^2 + 112x - 9 - 30x - 25x^2 = 0}}}
                                             {{{- 9x^2 + 82x - 9 = 0}}}        OR       {{{9x^2 - 82x + 9 = 0}}}
                                                                                            {{{9x^2 - 81x - x + 9 = 0}}}
                                                                                       9x(x - 9) - 1(x - 9) = 0
                                                                                               (x - 9)(9x - 1) = 0
                                                                                                x - 9 = 0               OR      9x - 1 = 0
                                                                                                      x = 0 + 9        OR           9x = 1
                                                                                                      x = 9               OR              x = {{{1/9}}} <font color = red><font size = 3>(IGNORE)</font></font>
{{{1/9}}} proves to be EXTRANEOUS, so sole solution is: <font color = red><font size = 3>x = 9</font></font>
To some though, it's easier to solve, if one of the left-side RADICALS is MOVED to the right, 1st. But then, this's subjective.


It looks exactly the same as before!! I'm just really confused by this problem, and I have a couple more like it, so I want
to know if figuring this one out could help me solve the other ones. <font color = red>Hopefully, the above will clear up SOME/ALL confusion!</font>

I'm confused about FOILing the different sides, whether or not I can combine two square roots, and quite frankly, a lot of
other things.  <font color = red>Hopefully, the above will clear up SOME/ALL confusion!</font>

One of the options on the test is 9, and I think this is the answer, because I've inserted it into the original equation
and it works, but I'm just confused about how to actually get 9 out of this.  <font color = red>Yes, the solution is indeed 9, as seen above! </font>

Sorry for the long question. I hope it's not hard to understand. I'm just hoping someone can walk me through all the steps
of solving a problem like this so I can do it easily in the future.  <font color = red>Hopefully, this author has assisted you in understanding this 
problem, so you can understand and obtain SOLUTIONS to similar problems, more easily, more efficiently, and without confusion!</font></font></font></font></b></pre>