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I am currently doing some questions for loci of complex numbers and this question stumped me.

I did some algebra, and got to here:

$$128(x^2 - 2x) + 144y^2 = 1024.$$

However, the answer scheme then simplifies this to:

$$128(x-1)^2 + 144y^2 = 1152.$$

How did this happen? Shouldn't the $1024$ become $1025$ instead (from the $(x-1)^2$ )?

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You have

$$\begin{equation}\begin{aligned} 128(x^2 - 2x) + 144y^2 & = 1024 \\ 128(x^2 - 2x + 1 - 1) + 144y^2 & = 1024 \\ 128(x^2 - 2x + 1) \color{red}{- 128} + 144y^2 & = 1024 \\ 128(x - 1)^2 + 144y^2 & = 1024 \color{red}{+ 128} \\ 128(x - 1)^2 + 144y^2 & = 1152 \end{aligned}\end{equation}\tag{1}\label{eq1A}$$

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No; don't forget about the factor $128$ before the term $(x-1)^2$. You get $$128(x-1)^2=128(x^2-2x+1)=128(x^2-2x)+128.$$

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