Solve by Factoring 2=-8/(x-6)+10/((x-6)^2)

$2 = \frac{- 8}{x - 6} + \frac{10}{{(x - 6)}^{2}}$

Step 1

Move all the expressions to the left side of the equation.

Subtract $\frac{- 8}{x - 6}$ from both sides of the equation.

$2 - \frac{- 8}{x - 6} = \frac{10}{{(x - 6)}^{2}}$

Subtract $\frac{10}{{(x - 6)}^{2}}$ from both sides of the equation.

$2 - \frac{- 8}{x - 6} - \frac{10}{{(x - 6)}^{2}} = 0$

Step 2

Simplify $2 - \frac{- 8}{x - 6} - \frac{10}{{(x - 6)}^{2}}$ .

Find the common denominator.

Write $2$ as a fraction with denominator $1$ .

$\frac{2}{1} - \frac{- 8}{x - 6} - \frac{10}{{(x - 6)}^{2}} = 0$

Multiply $\frac{2}{1}$ by $\frac{{(x - 6)}^{2}}{{(x - 6)}^{2}}$ .

$\frac{2}{1} \cdot \frac{{(x - 6)}^{2}}{{(x - 6)}^{2}} - \frac{- 8}{x - 6} - \frac{10}{{(x - 6)}^{2}} = 0$

Multiply $\frac{2}{1}$ by $\frac{{(x - 6)}^{2}}{{(x - 6)}^{2}}$ .

$\frac{2 {(x - 6)}^{2}}{{(x - 6)}^{2}} - \frac{- 8}{x - 6} - \frac{10}{{(x - 6)}^{2}} = 0$

Multiply $\frac{- 8}{x - 6}$ by $\frac{x - 6}{x - 6}$ .

$\frac{2 {(x - 6)}^{2}}{{(x - 6)}^{2}} - (\frac{- 8}{x - 6} \cdot \frac{x - 6}{x - 6}) - \frac{10}{{(x - 6)}^{2}} = 0$

Multiply $\frac{- 8}{x - 6}$ by $\frac{x - 6}{x - 6}$ .

$\frac{2 {(x - 6)}^{2}}{{(x - 6)}^{2}} - \frac{- 8 (x - 6)}{(x - 6) (x - 6)} - \frac{10}{{(x - 6)}^{2}} = 0$

Raise $x - 6$ to the power of $1$ .

$\frac{2 {(x - 6)}^{2}}{{(x - 6)}^{2}} - \frac{- 8 (x - 6)}{{(x - 6)}^{1} (x - 6)} - \frac{10}{{(x - 6)}^{2}} = 0$

Raise $x - 6$ to the power of $1$ .

$\frac{2 {(x - 6)}^{2}}{{(x - 6)}^{2}} - \frac{- 8 (x - 6)}{{(x - 6)}^{1} {(x - 6)}^{1}} - \frac{10}{{(x - 6)}^{2}} = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 {(x - 6)}^{2}}{{(x - 6)}^{2}} - \frac{- 8 (x - 6)}{{(x - 6)}^{1 + 1}} - \frac{10}{{(x - 6)}^{2}} = 0$

Add $1$ and $1$ .

$\frac{2 {(x - 6)}^{2}}{{(x - 6)}^{2}} - \frac{- 8 (x - 6)}{{(x - 6)}^{2}} - \frac{10}{{(x - 6)}^{2}} = 0$

Combine the numerators over the common denominator.

$\frac{2 {(x - 6)}^{2} + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Simplify each term.

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$\frac{2 ((x - 6) (x - 6)) + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Expand $(x - 6) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 (x (x - 6) - 6 (x - 6)) + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Apply the distributive property.

$\frac{2 (x \cdot x + x \cdot - 6 - 6 (x - 6)) + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Apply the distributive property.

$\frac{2 (x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6) + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{2 (x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6) + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Move $- 6$ to the left of $x$ .

$\frac{2 (x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6) + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Multiply $- 6$ by $- 6$ .

$\frac{2 (x^{2} - 6 x - 6 x + 36) + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Subtract $6 x$ from $- 6 x$ .

$\frac{2 (x^{2} - 12 x + 36) + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Apply the distributive property.

$\frac{2 x^{2} + 2 (- 12 x) + 2 \cdot 36 + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Simplify.

Multiply $- 12$ by $2$ .

$\frac{2 x^{2} - 24 x + 2 \cdot 36 + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Multiply $2$ by $36$ .

$\frac{2 x^{2} - 24 x + 72 + 8 (x - 6) - 10}{{(x - 6)}^{2}} = 0$

Apply the distributive property.

$\frac{2 x^{2} - 24 x + 72 + 8 x + 8 \cdot - 6 - 10}{{(x - 6)}^{2}} = 0$

Multiply $8$ by $- 6$ .

$\frac{2 x^{2} - 24 x + 72 + 8 x - 48 - 10}{{(x - 6)}^{2}} = 0$

Add $- 24 x$ and $8 x$ .

$\frac{2 x^{2} - 16 x + 72 - 48 - 10}{{(x - 6)}^{2}} = 0$

Subtract $48$ from $72$ .

$\frac{2 x^{2} - 16 x + 24 - 10}{{(x - 6)}^{2}} = 0$

Subtract $10$ from $24$ .

$\frac{2 x^{2} - 16 x + 14}{{(x - 6)}^{2}} = 0$

Simplify the numerator.

Factor $2$ out of $2 x^{2} - 16 x + 14$ .

Factor $2$ out of $2 x^{2}$ .

$\frac{2 (x^{2}) - 16 x + 14}{{(x - 6)}^{2}} = 0$

Factor $2$ out of $- 16 x$ .

$\frac{2 (x^{2}) + 2 (- 8 x) + 14}{{(x - 6)}^{2}} = 0$

Factor $2$ out of $14$ .

$\frac{2 x^{2} + 2 (- 8 x) + 2 \cdot 7}{{(x - 6)}^{2}} = 0$

Factor $2$ out of $2 x^{2} + 2 (- 8 x)$ .

$\frac{2 (x^{2} - 8 x) + 2 \cdot 7}{{(x - 6)}^{2}} = 0$

Factor $2$ out of $2 (x^{2} - 8 x) + 2 \cdot 7$ .

$\frac{2 (x^{2} - 8 x + 7)}{{(x - 6)}^{2}} = 0$

Factor $x^{2} - 8 x + 7$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $7$ and whose sum is $- 8$ .

$- 7, - 1$

Write the factored form using these integers.

$\frac{2 ((x - 7) (x - 1))}{{(x - 6)}^{2}} = 0$

$\frac{2 (x - 7) (x - 1)}{{(x - 6)}^{2}} = 0$

Step 3

Set the numerator equal to zero.

$2 (x - 7) (x - 1) = 0$

Step 4

Solve the equation for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 7 = 0$

$x - 1 = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $2 (x - 7) (x - 1) = 0$ true.

$x = 7, 1$

Solve by Factoring 2=-8/(x-6)+10/((x-6)^2)

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