A student is asked to solve
and gives the following solution.
Solution:
<---Here is the error!
This is the false assumption that "the square of the sum"
is the same as "the sum of the squares". But it is not!
Never confuse these two!
Here is the correct solution:
Isolate one of the radicals. It doesn't matter which one. I'll isolate
the first one by subtracting the second one from both sides:
Indicate the squaring of both sides:
Simplify the left side by eliminating the square root and the 2 exponent.
Rewrite the right side as the binomial times itself:
Next we use FOIL on the right side:
F O I L
| | | |
Combine the middle two radical terms on the right,
and simplify the first and last terms on the right:
Isolate the term with the radical on the left side
Divide every term by 2
Square both sides:
Get 0 on the left:
Swap sides:
This does not factor so we must use the quadratic formula:
Write 128 as 64*2
Simplify the radical:
Factor 2 out of the numerator:
Cancel the factors of 2:
We must check every radical equation for extraneous answers.
It is very difficult to check radicals inside radicals, and this
makes me wonder if you copied the problem right since this one
comes out so messy. But I'll assume that you did.
We must check both solutions and
However to make things a little easier with these terrible answers,
we will use their decimal approximations,
Checking
That does not check so is an extraneous solution.
----
Checking
That is a very close check so is
the only solution to the original.
Edwin