SOLUTION: Please solve A. (3-√(x+2) )^2=4 and B. 3+5√(1+2x ^2 )=13 thank you!

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Question 177767: Please solve
A. (3-√(x+2) )^2=4
and
B. 3+5√(1+2x ^2 )=13

thank you!
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!


A.

Step 1: Expand the binomial on the left:



Step 2: A simple substitution will allow us to deal with this ugliness in a fairly straightforward manner.

Let , then:



Step 3: Put the quadratic into standard form:



Giving us a factorable quadratic in t, so:

Step 4: Factor the quadratic:



Therefore or

But we aren't done yet.

Step 5: Since ,







But we are still not done.

Step 6: Check the potential roots to ensure that we have not introduced extraneous roots in the process of solving the problem:

If then,






True statement, potential solution confirmed.

If then,








True statement, potential solution confirmed.

Now we are done.

=======================================================================
B.

Step 1: Add -3 to both sides and then multiply by to obtain:



Step 2: Square both sides:



Step 3: Add -1 to both sides and then multiply by to obtain:



Step 4: Take the square root of both sides, considering both the positive and negative roots, and then rationalize the denominators:





Step 5: Check the potential solutions.

Since regardless of whether or , if one solution is good, then they both are:








True statement, both potential solutions confirmed.
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