Solve 3x+A^4/9+A^2=9-A^2

$$3 x + \frac { A ^ { 4 } } { 9 + A ^ { 2 } } = 9 - A ^ { 2 }$$

Solve for x (complex solution)

$$x=-\frac{2A^{4}-81}{3\left(A^{2}+9\right)}$$

$$A\neq -3i\text{ and }A\neq 3i$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(A-3i\right)\left(A+3i\right).

$$3x\left(A-3i\right)\left(A+3i\right)+A^{4}=\left(A-3i\right)\left(A+3i\right)\times 9-A^{2}\left(A-3i\right)\left(A+3i\right)$$

Use the distributive property to multiply 3x by A-3i.

$$\left(3xA-9ix\right)\left(A+3i\right)+A^{4}=\left(A-3i\right)\left(A+3i\right)\times 9-A^{2}\left(A-3i\right)\left(A+3i\right)$$

Use the distributive property to multiply 3xA-9ix by A+3i and combine like terms.

$$3xA^{2}+27x+A^{4}=\left(A-3i\right)\left(A+3i\right)\times 9-A^{2}\left(A-3i\right)\left(A+3i\right)$$

Use the distributive property to multiply A-3i by A+3i and combine like terms.

$$3xA^{2}+27x+A^{4}=\left(A^{2}+9\right)\times 9-A^{2}\left(A-3i\right)\left(A+3i\right)$$

Use the distributive property to multiply A^{2}+9 by 9.

$$3xA^{2}+27x+A^{4}=9A^{2}+81-A^{2}\left(A-3i\right)\left(A+3i\right)$$

Use the distributive property to multiply -A^{2} by A-3i.

$$3xA^{2}+27x+A^{4}=9A^{2}+81+\left(-A^{3}+3iA^{2}\right)\left(A+3i\right)$$

Use the distributive property to multiply -A^{3}+3iA^{2} by A+3i and combine like terms.

$$3xA^{2}+27x+A^{4}=9A^{2}+81-A^{4}-9A^{2}$$

Combine 9A^{2} and -9A^{2} to get 0.

$$3xA^{2}+27x+A^{4}=81-A^{4}$$

Subtract A^{4} from both sides.

$$3xA^{2}+27x=81-A^{4}-A^{4}$$

Combine -A^{4} and -A^{4} to get -2A^{4}.

$$3xA^{2}+27x=81-2A^{4}$$

Combine all terms containing x.

$$\left(3A^{2}+27\right)x=81-2A^{4}$$

Divide both sides by 3A^{2}+27.

$$\frac{\left(3A^{2}+27\right)x}{3A^{2}+27}=\frac{81-2A^{4}}{3A^{2}+27}$$

Dividing by 3A^{2}+27 undoes the multiplication by 3A^{2}+27.

$$x=\frac{81-2A^{4}}{3A^{2}+27}$$

Divide 81-2A^{4} by 3A^{2}+27.

$$x=\frac{81-2A^{4}}{3\left(A^{2}+9\right)}$$

Solve for x

$$x=-\frac{2A^{4}-81}{3\left(A^{2}+9\right)}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by A^{2}+9.

$$3x\left(A^{2}+9\right)+A^{4}=\left(A^{2}+9\right)\times 9-A^{2}\left(A^{2}+9\right)$$

Use the distributive property to multiply 3x by A^{2}+9.

$$3xA^{2}+27x+A^{4}=\left(A^{2}+9\right)\times 9-A^{2}\left(A^{2}+9\right)$$

Use the distributive property to multiply A^{2}+9 by 9.

$$3xA^{2}+27x+A^{4}=9A^{2}+81-A^{2}\left(A^{2}+9\right)$$

Use the distributive property to multiply -A^{2} by A^{2}+9.

$$3xA^{2}+27x+A^{4}=9A^{2}+81-A^{4}-9A^{2}$$

Combine 9A^{2} and -9A^{2} to get 0.

$$3xA^{2}+27x+A^{4}=81-A^{4}$$

Subtract A^{4} from both sides.

$$3xA^{2}+27x=81-A^{4}-A^{4}$$

Combine -A^{4} and -A^{4} to get -2A^{4}.

$$3xA^{2}+27x=81-2A^{4}$$

Combine all terms containing x.

$$\left(3A^{2}+27\right)x=81-2A^{4}$$

Divide both sides by 3A^{2}+27.

$$\frac{\left(3A^{2}+27\right)x}{3A^{2}+27}=\frac{81-2A^{4}}{3A^{2}+27}$$

Dividing by 3A^{2}+27 undoes the multiplication by 3A^{2}+27.

$$x=\frac{81-2A^{4}}{3A^{2}+27}$$

Divide 81-2A^{4} by 3A^{2}+27.

$$x=\frac{81-2A^{4}}{3\left(A^{2}+9\right)}$$

Solve for A (complex solution)

$$A=\frac{\sqrt{3\sqrt{x^{2}-24x+72}-3x}}{2}$$

$$A=-\frac{\sqrt{3\sqrt{x^{2}-24x+72}-3x}}{2}$$

$$A=-\frac{\sqrt{-3\sqrt{x^{2}-24x+72}-3x}}{2}$$

$$A=\frac{\sqrt{-3\sqrt{x^{2}-24x+72}-3x}}{2}$$

Solve for A

$$A=-\frac{\sqrt{3\left(\sqrt{x^{2}-24x+72}-x\right)}}{2}$$

$$A=\frac{\sqrt{3\left(\sqrt{x^{2}-24x+72}-x\right)}}{2}\text{, }x\leq 3$$