Solve (a^3+8):(a^2-4)=x:(a-2)

$$( a ^ { 3 } + 8 ) : ( a ^ { 2 } - 4 ) = x : ( a - 2 )$$

Solve for x (complex solution)

$$x=a^{2}-2a+4$$

$$a\neq -2\text{ and }a\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(a-2\right)\left(a+2\right), the least common multiple of a^{2}-4,a-2.

$$a^{3}+8=\left(a+2\right)x$$

Use the distributive property to multiply a+2 by x.

$$a^{3}+8=ax+2x$$

Swap sides so that all variable terms are on the left hand side.

$$ax+2x=a^{3}+8$$

Combine all terms containing x.

$$\left(a+2\right)x=a^{3}+8$$

Divide both sides by a+2.

$$\frac{\left(a+2\right)x}{a+2}=\frac{a^{3}+8}{a+2}$$

Dividing by a+2 undoes the multiplication by a+2.

$$x=\frac{a^{3}+8}{a+2}$$

Divide a^{3}+8 by a+2.

$$x=a^{2}-2a+4$$

Solve for x

$$x=a^{2}-2a+4$$

$$|a|\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(a-2\right)\left(a+2\right), the least common multiple of a^{2}-4,a-2.

$$a^{3}+8=\left(a+2\right)x$$

Use the distributive property to multiply a+2 by x.

$$a^{3}+8=ax+2x$$

Swap sides so that all variable terms are on the left hand side.

$$ax+2x=a^{3}+8$$

Combine all terms containing x.

$$\left(a+2\right)x=a^{3}+8$$

Divide both sides by a+2.

$$\frac{\left(a+2\right)x}{a+2}=\frac{a^{3}+8}{a+2}$$

Dividing by a+2 undoes the multiplication by a+2.

$$x=\frac{a^{3}+8}{a+2}$$

Divide a^{3}+8 by a+2.

$$x=a^{2}-2a+4$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\sqrt{x-3}+1\text{, }&x\neq 12\\a=\sqrt{x-3}+1\text{, }&x\neq 4\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=-\sqrt{x-3}+1\text{, }&x\geq 3\text{ and }x\neq 12\\a=\sqrt{x-3}+1\text{, }&x\neq 4\text{ and }x\geq 3\end{matrix}\right.$$