Complete the square on $\frac{x^2+y^2-2xy+4kty}{4kt}$

Question

In solving a problem, my lecturer has informed us that the we need to complete the square on $\frac{x^2+y^2-2xy+4kty}{4kt}$.

The solution to this problem is that $-\frac{(y+2kt-x)^2}{4kt}-x+kt.$ But I really do not understand how this is achieved, can anyone explain how this is done?

Notice that : $$(a + b + c)^2 = a^2 + b^2 + c^2 + 2 a b + 2 a c + 2 b c$$ — Essaidi, Jan 03 '23 at 13:45

score 0 · Answer 1 · answered Jan 03 '23 at 13:58

Hint: Write $x^2+y^2-2xy+4kty=x^2+(-y)^2+(-2kt)^2+2(x)(-y)+2(x)(-2kt)+2(-y)(-2kt)-4k^2t^2+4xkt=(x-y-2kt)^2-4k^2t^2+4xkt$. When you divide this by $4kt$ you should get the desire expression above.Also I think the solution you quoted is not correct. It should be minus of the solution.

Complete the square on $\frac{x^2+y^2-2xy+4kty}{4kt}$

1 Answers1