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

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

Simplify $\frac{6 k^{2} - 36 k}{A} \cdot \frac{k^{2} + 12 k + 36}{k^{2} - 36}$ .

Factor $6 k$ out of $6 k^{2} - 36 k$ .

Factor $6 k$ out of $6 k^{2}$ .

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

Factor $6 k$ out of $- 36 k$ .

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

Factor $6 k$ out of $6 k (k) + 6 k (- 6)$ .

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

Factor using the perfect square rule.

Rewrite $36$ as $6^{2}$ .

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 k = 2 \cdot k \cdot 6$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = k$ and $b = 6$ .

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

Simplify the denominator.

Rewrite $36$ as $6^{2}$ .

$\frac{6 k (k - 6)}{A} \cdot \frac{{(k + 6)}^{2}}{k^{2} - 6^{2}} = 1$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = k$ and $b = 6$ .

$\frac{6 k (k - 6)}{A} \cdot \frac{{(k + 6)}^{2}}{(k + 6) (k - 6)} = 1$

Simplify terms.

Combine.

$\frac{6 k (k - 6) {(k + 6)}^{2}}{A ((k + 6) (k - 6))} = 1$

Cancel the common factor of $k - 6$ .

Cancel the common factor.

$\frac{6 k (k - 6) {(k + 6)}^{2}}{A ((k + 6) (k - 6))} = 1$

Rewrite the expression.

$\frac{6 k {(k + 6)}^{2}}{A (k + 6)} = 1$

Cancel the common factor of ${(k + 6)}^{2}$ and $k + 6$ .

Factor $k + 6$ out of $6 k {(k + 6)}^{2}$ .

$\frac{(k + 6) (6 k (k + 6))}{A (k + 6)} = 1$

Cancel the common factors.

Factor $k + 6$ out of $A (k + 6)$ .

$\frac{(k + 6) (6 k (k + 6))}{(k + 6) A} = 1$

Cancel the common factor.

$\frac{(k + 6) (6 k (k + 6))}{(k + 6) A} = 1$

Rewrite the expression.

$\frac{6 k (k + 6)}{A} = 1$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$A, 1$

The LCM of one and any expression is the expression.

$A$

Multiply each term in $\frac{6 k (k + 6)}{A} = 1$ by $A$ to eliminate the fractions.

Multiply each term in $\frac{6 k (k + 6)}{A} = 1$ by $A$ .

$\frac{6 k (k + 6)}{A} A = 1 A$

Simplify the left side.

Cancel the common factor of $A$ .

Cancel the common factor.

$\frac{6 k (k + 6)}{A} A = 1 A$

Rewrite the expression.

$6 k (k + 6) = 1 A$

Apply the distributive property.

$6 k \cdot k + 6 k \cdot 6 = 1 A$

Multiply $k$ by $k$ by adding the exponents.

Move $k$ .

$6 (k \cdot k) + 6 k \cdot 6 = 1 A$

Multiply $k$ by $k$ .

$6 k^{2} + 6 k \cdot 6 = 1 A$

Multiply $6$ by $6$ .

$6 k^{2} + 36 k = 1 A$

Simplify the right side.

Multiply $A$ by $1$ .

$6 k^{2} + 36 k = A$

Rewrite the equation as $A = 6 k^{2} + 36 k$ .

$A = 6 k^{2} + 36 k$

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

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