Solve for x
x=-\frac{2A^{4}-81}{3\left(A^{2}+9\right)}
Solve for A
A=-\frac{\sqrt{3\left(\sqrt{x^{2}-24x+72}-x\right)}}{2}
A=\frac{\sqrt{3\left(\sqrt{x^{2}-24x+72}-x\right)}}{2}\text{, }x\leq 3
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3x\left(A^{2}+9\right)+A^{4}=\left(A^{2}+9\right)\times 9-A^{2}\left(A^{2}+9\right)
Multiply both sides of the equation by A^{2}+9.
3xA^{2}+27x+A^{4}=\left(A^{2}+9\right)\times 9-A^{2}\left(A^{2}+9\right)
Use the distributive property to multiply 3x by A^{2}+9.
3xA^{2}+27x+A^{4}=9A^{2}+81-A^{2}\left(A^{2}+9\right)
Use the distributive property to multiply A^{2}+9 by 9.
3xA^{2}+27x+A^{4}=9A^{2}+81-A^{4}-9A^{2}
Use the distributive property to multiply -A^{2} by A^{2}+9.
3xA^{2}+27x+A^{4}=81-A^{4}
Combine 9A^{2} and -9A^{2} to get 0.
3xA^{2}+27x=81-A^{4}-A^{4}
Subtract A^{4} from both sides.
3xA^{2}+27x=81-2A^{4}
Combine -A^{4} and -A^{4} to get -2A^{4}.
\left(3A^{2}+27\right)x=81-2A^{4}
Combine all terms containing x.
\frac{\left(3A^{2}+27\right)x}{3A^{2}+27}=\frac{81-2A^{4}}{3A^{2}+27}
Divide both sides by 3A^{2}+27.
x=\frac{81-2A^{4}}{3A^{2}+27}
Dividing by 3A^{2}+27 undoes the multiplication by 3A^{2}+27.
x=\frac{81-2A^{4}}{3\left(A^{2}+9\right)}
Divide 81-2A^{4} by 3A^{2}+27.
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