Misstep With Discriminant and equations writable as quadratic form

Question

I missed a step in my equation and would like to know what i'm doing wrong.

I have the following equation: ${x^{4} - {\color{red}15} x^{2} + {\color{red}54} = 0}$

Now, we let ${y = x^2}$ We can now rewrite our equation to... $${y^2-15y+54}$$ We can also factor it into 2 binomials.. $${(y-6)(y-9)}$$

Here is where I am stuck however, My answer after that is $$\sqrt{-6},\sqrt{6},\sqrt{-9}.\sqrt{9}$$

This certianly is not the answer as I am provided 4 options, here is one just for reference (May not be the correct solution) $$x=3,-3, \sqrt{y}, -\sqrt{6} $$

Thanks!

Answer 1

Your factorization is incorrect. It should be $$ y^2−15y+54 = (y-6)(y-9) $$

Note that your factorization yields this: $$ (y-18)(y+3)=y^2−15y{\color{red}-}54 $$