Solve 6k^2-36k/A*k^2+12k+36/k^2-36=1

$$\frac { 6 k ^ { 2 } - 36 k } { A } \cdot \frac { k ^ { 2 } + 12 k + 36 } { k ^ { 2 } - 36 } = 1$$

Solve for A

$$A=6k\left(k+6\right)$$

$$k\neq 0\text{ and }|k|\neq 6$$

Steps for Solving Linear Equation

Variable A cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by A\left(k-6\right)\left(k+6\right), the least common multiple of A,k^{2}-36.

$$\left(k^{2}-36\right)\left(6k^{2}-36k\right)\times \frac{k^{2}+12k+36}{k^{2}-36}=A\left(k-6\right)\left(k+6\right)$$

Factor the expressions that are not already factored in \frac{k^{2}+12k+36}{k^{2}-36}.

$$\left(k^{2}-36\right)\left(6k^{2}-36k\right)\times \frac{\left(k+6\right)^{2}}{\left(k-6\right)\left(k+6\right)}=A\left(k-6\right)\left(k+6\right)$$

Cancel out k+6 in both numerator and denominator.

$$\left(k^{2}-36\right)\left(6k^{2}-36k\right)\times \frac{k+6}{k-6}=A\left(k-6\right)\left(k+6\right)$$

Express \left(k^{2}-36\right)\times \frac{k+6}{k-6} as a single fraction.

$$\frac{\left(k^{2}-36\right)\left(k+6\right)}{k-6}\left(6k^{2}-36k\right)=A\left(k-6\right)\left(k+6\right)$$

Use the distributive property to multiply \frac{\left(k^{2}-36\right)\left(k+6\right)}{k-6} by 6k^{2}-36k.

$$6\times \frac{\left(k^{2}-36\right)\left(k+6\right)}{k-6}k^{2}-36\times \frac{\left(k^{2}-36\right)\left(k+6\right)}{k-6}k=A\left(k-6\right)\left(k+6\right)$$

Factor the expressions that are not already factored in \frac{\left(k^{2}-36\right)\left(k+6\right)}{k-6}.

$$6\times \frac{\left(k-6\right)\left(k+6\right)^{2}}{k-6}k^{2}-36\times \frac{\left(k^{2}-36\right)\left(k+6\right)}{k-6}k=A\left(k-6\right)\left(k+6\right)$$

Cancel out k-6 in both numerator and denominator.

$$6\left(k+6\right)^{2}k^{2}-36\times \frac{\left(k^{2}-36\right)\left(k+6\right)}{k-6}k=A\left(k-6\right)\left(k+6\right)$$

Expand the expression.

$$6\left(k^{2}+12k+36\right)k^{2}-36\times \frac{\left(k^{2}-36\right)\left(k+6\right)}{k-6}k=A\left(k-6\right)\left(k+6\right)$$

Use the distributive property to multiply 6 by k^{2}+12k+36.

$$\left(6k^{2}+72k+216\right)k^{2}-36\times \frac{\left(k^{2}-36\right)\left(k+6\right)}{k-6}k=A\left(k-6\right)\left(k+6\right)$$

Use the distributive property to multiply 6k^{2}+72k+216 by k^{2}.

$$6k^{4}+72k^{3}+216k^{2}-36\times \frac{\left(k^{2}-36\right)\left(k+6\right)}{k-6}k=A\left(k-6\right)\left(k+6\right)$$

Factor the expressions that are not already factored in \frac{\left(k^{2}-36\right)\left(k+6\right)}{k-6}.

$$6k^{4}+72k^{3}+216k^{2}-36\times \frac{\left(k-6\right)\left(k+6\right)^{2}}{k-6}k=A\left(k-6\right)\left(k+6\right)$$

Cancel out k-6 in both numerator and denominator.

$$6k^{4}+72k^{3}+216k^{2}-36\left(k+6\right)^{2}k=A\left(k-6\right)\left(k+6\right)$$

Expand the expression.

$$6k^{4}+72k^{3}+216k^{2}-36\left(k^{2}+12k+36\right)k=A\left(k-6\right)\left(k+6\right)$$

Use the distributive property to multiply -36 by k^{2}+12k+36.

$$6k^{4}+72k^{3}+216k^{2}+\left(-36k^{2}-432k-1296\right)k=A\left(k-6\right)\left(k+6\right)$$

Use the distributive property to multiply -36k^{2}-432k-1296 by k.

$$6k^{4}+72k^{3}+216k^{2}-36k^{3}-432k^{2}-1296k=A\left(k-6\right)\left(k+6\right)$$

Combine 72k^{3} and -36k^{3} to get 36k^{3}.

$$6k^{4}+36k^{3}+216k^{2}-432k^{2}-1296k=A\left(k-6\right)\left(k+6\right)$$

Combine 216k^{2} and -432k^{2} to get -216k^{2}.

$$6k^{4}+36k^{3}-216k^{2}-1296k=A\left(k-6\right)\left(k+6\right)$$

Use the distributive property to multiply A by k-6.

$$6k^{4}+36k^{3}-216k^{2}-1296k=\left(Ak-6A\right)\left(k+6\right)$$

Use the distributive property to multiply Ak-6A by k+6 and combine like terms.

$$6k^{4}+36k^{3}-216k^{2}-1296k=Ak^{2}-36A$$

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

$$Ak^{2}-36A=6k^{4}+36k^{3}-216k^{2}-1296k$$

Combine all terms containing A.

$$\left(k^{2}-36\right)A=6k^{4}+36k^{3}-216k^{2}-1296k$$

Divide both sides by k^{2}-36.

$$\frac{\left(k^{2}-36\right)A}{k^{2}-36}=\frac{6k\left(k-6\right)\left(k+6\right)^{2}}{k^{2}-36}$$

Dividing by k^{2}-36 undoes the multiplication by k^{2}-36.

$$A=\frac{6k\left(k-6\right)\left(k+6\right)^{2}}{k^{2}-36}$$

Divide 6k\left(-6+k\right)\left(6+k\right)^{2} by k^{2}-36.

$$A=6k\left(k+6\right)$$

Variable A cannot be equal to 0.

$$A=6k\left(k+6\right)\text{, }A\neq 0$$

Solve for k

$$\left\{\begin{matrix}k=-\frac{\sqrt{6A+324}}{6}-3\text{, }&A\neq 0\text{ and }A\geq -54\\k=\frac{\sqrt{6A+324}}{6}-3\text{, }&A\neq 432\text{ and }A\geq -54\text{ and }A\neq 0\end{matrix}\right.$$