First, starting from the equation:
[tex]y=x^2+12x[/tex]
We should change every x for a y and every y for an x, so:
[tex]x=y^2+12y[/tex]
In the text, the first step was:
[tex]x=\questeq y^2+12x[/tex]
So, the first mistake was not to change the linear term 12x for 12y.
Another mistake was made from the following steps:
[tex]\begin{gathered} y^2=-11x \\ y=\sqrt[]{-11x},x\ge0 \end{gathered}[/tex]
The expression inside the square root must be greater or equal to 0, then:
[tex]\begin{gathered} -11x\ge0 \\ \Rightarrow x\le0 \end{gathered}[/tex]
So, the second mistake was not to identify the domain of x correctly.
A third mistake is that the square root of y^2 is not y, it is |y|. So:
[tex]\begin{gathered} y^2=-11x \\ \Rightarrow|y|=\sqrt[]{-11x} \\ \Rightarrow y=\pm\sqrt[]{-11x} \end{gathered}[/tex]
