Hello, p.numminen!
How would you calculate: . ?
There is a very sneaky trick we can pull . . .
First, we notice that: . .just happens to equal . . Check it out!
Then: .
So the problem becomes: .
We can simplify by simultaneous equations as well.
Let
Squaring, we get
Now by matching up coefficients, we get two equations:
...[1]
...[2]
...[2']
Now substituting into [1]:
Now we make the substitution , then it becomes:
or
Then or
So or
So the solutions are \sqrt{5},\frac{1}{\sqrt{5}}),(-\sqrt{5},-\frac{1}{\sqrt{5}}),(\sqrt{2},\frac{1}{\sqrt{2}}), (-\sqrt{2},-\frac{1}{\sqrt{2}})\}" alt="\{(A,B)\sqrt{5},\frac{1}{\sqrt{5}}),(-\sqrt{5},-\frac{1}{\sqrt{5}}),(\sqrt{2},\frac{1}{\sqrt{2}}), (-\sqrt{2},-\frac{1}{\sqrt{2}})\}" />
Let's try the first one; then we have:
You can check this by equating both sides as Soroban did. Note that , also works, but the other two solutions Do Not. They are the negative solutions. That's why you have to check the final result to see if it fits!
Of course it then follows that