\frac { 1 } { ( \sqrt { 2 } + 2 \sqrt { 3 } ) ^ { 2 } }
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The answer above is completely correct, but it may help the student to see how to reach it:
We seek a square root of 3−2√2. This must have the form a+b√2.
(a+b√2)2=a2+2ab√2+2b2
So
a2+2b2=3
and
2ab=−2⇒ab=−1
These are two simultaneous equations for a and b.
Rearrange the second:
b=−1a
Substitute into the first:
a2+2a2=3
a4+2=3a2
a4−3a2+2=0
Note that this is a quadratic in a2:
(a2)2−3(a2)+2=0
Factorise:
(a2−2)(a2−1)=0
This gives us two possible solutions for a2: 2 and 1, and so the four solutions for a: ±
Explanation:
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