Solve by Factoring 3/(x-7)+4/(x+2)=1

$\frac{3}{x - 7} + \frac{4}{x + 2} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{3}{x - 7} + \frac{4}{x + 2} - 1 = 0$

Step 2

Simplify $\frac{3}{x - 7} + \frac{4}{x + 2} - 1$ .

To write $\frac{3}{x - 7}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{3}{x - 7} \cdot \frac{x + 2}{x + 2} + \frac{4}{x + 2} - 1 = 0$

To write $\frac{4}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$\frac{3}{x - 7} \cdot \frac{x + 2}{x + 2} + \frac{4}{x + 2} \cdot \frac{x - 7}{x - 7} - 1 = 0$

Write each expression with a common denominator of $(x - 7) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{x - 7}$ by $\frac{x + 2}{x + 2}$ .

$\frac{3 (x + 2)}{(x - 7) (x + 2)} + \frac{4}{x + 2} \cdot \frac{x - 7}{x - 7} - 1 = 0$

Multiply $\frac{4}{x + 2}$ by $\frac{x - 7}{x - 7}$ .

$\frac{3 (x + 2)}{(x - 7) (x + 2)} + \frac{4 (x - 7)}{(x + 2) (x - 7)} - 1 = 0$

Reorder the factors of $(x - 7) (x + 2)$ .

$\frac{3 (x + 2)}{(x + 2) (x - 7)} + \frac{4 (x - 7)}{(x + 2) (x - 7)} - 1 = 0$

Combine the numerators over the common denominator.

$\frac{3 (x + 2) + 4 (x - 7)}{(x + 2) (x - 7)} - 1 = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{3 x + 3 \cdot 2 + 4 (x - 7)}{(x + 2) (x - 7)} - 1 = 0$

Multiply $3$ by $2$ .

$\frac{3 x + 6 + 4 (x - 7)}{(x + 2) (x - 7)} - 1 = 0$

Apply the distributive property.

$\frac{3 x + 6 + 4 x + 4 \cdot - 7}{(x + 2) (x - 7)} - 1 = 0$

Multiply $4$ by $- 7$ .

$\frac{3 x + 6 + 4 x - 28}{(x + 2) (x - 7)} - 1 = 0$

Add $3 x$ and $4 x$ .

$\frac{7 x + 6 - 28}{(x + 2) (x - 7)} - 1 = 0$

Subtract $28$ from $6$ .

$\frac{7 x - 22}{(x + 2) (x - 7)} - 1 = 0$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{(x + 2) (x - 7)}{(x + 2) (x - 7)}$ .

$\frac{7 x - 22}{(x + 2) (x - 7)} - 1 \cdot \frac{(x + 2) (x - 7)}{(x + 2) (x - 7)} = 0$

Combine $- 1$ and $\frac{(x + 2) (x - 7)}{(x + 2) (x - 7)}$ .

$\frac{7 x - 22}{(x + 2) (x - 7)} + \frac{- ((x + 2) (x - 7))}{(x + 2) (x - 7)} = 0$

Combine the numerators over the common denominator.

$\frac{7 x - 22 - ((x + 2) (x - 7))}{(x + 2) (x - 7)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{7 x - 22 + (- x - 1 \cdot 2) (x - 7)}{(x + 2) (x - 7)} = 0$

Multiply $- 1$ by $2$ .

$\frac{7 x - 22 + (- x - 2) (x - 7)}{(x + 2) (x - 7)} = 0$

Expand $(- x - 2) (x - 7)$ using the FOIL Method.

Apply the distributive property.

$\frac{7 x - 22 - x (x - 7) - 2 (x - 7)}{(x + 2) (x - 7)} = 0$

Apply the distributive property.

$\frac{7 x - 22 - x \cdot x - x \cdot - 7 - 2 (x - 7)}{(x + 2) (x - 7)} = 0$

Apply the distributive property.

$\frac{7 x - 22 - x \cdot x - x \cdot - 7 - 2 x - 2 \cdot - 7}{(x + 2) (x - 7)} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{7 x - 22 - (x \cdot x) - x \cdot - 7 - 2 x - 2 \cdot - 7}{(x + 2) (x - 7)} = 0$

Multiply $x$ by $x$ .

$\frac{7 x - 22 - x^{2} - x \cdot - 7 - 2 x - 2 \cdot - 7}{(x + 2) (x - 7)} = 0$

Multiply $- 7$ by $- 1$ .

$\frac{7 x - 22 - x^{2} + 7 x - 2 x - 2 \cdot - 7}{(x + 2) (x - 7)} = 0$

Multiply $- 2$ by $- 7$ .

$\frac{7 x - 22 - x^{2} + 7 x - 2 x + 14}{(x + 2) (x - 7)} = 0$

Subtract $2 x$ from $7 x$ .

$\frac{7 x - 22 - x^{2} + 5 x + 14}{(x + 2) (x - 7)} = 0$

Add $7 x$ and $5 x$ .

$\frac{12 x - 22 - x^{2} + 14}{(x + 2) (x - 7)} = 0$

Add $- 22$ and $14$ .

$\frac{12 x - x^{2} - 8}{(x + 2) (x - 7)} = 0$

Reorder terms.

$\frac{- x^{2} + 12 x - 8}{(x + 2) (x - 7)} = 0$

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) + 12 x - 8}{(x + 2) (x - 7)} = 0$

Factor $- 1$ out of $12 x$ .

$\frac{- (x^{2}) - (- 12 x) - 8}{(x + 2) (x - 7)} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 12 x)$ .

$\frac{- (x^{2} - 12 x) - 8}{(x + 2) (x - 7)} = 0$

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{- (x^{2} - 12 x) - 1 (8)}{(x + 2) (x - 7)} = 0$

Factor $- 1$ out of $- (x^{2} - 12 x) - 1 (8)$ .

$\frac{- (x^{2} - 12 x + 8)}{(x + 2) (x - 7)} = 0$

Rewrite $- (x^{2} - 12 x + 8)$ as $- 1 (x^{2} - 12 x + 8)$ .

$\frac{- 1 (x^{2} - 12 x + 8)}{(x + 2) (x - 7)} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 12 x + 8}{(x + 2) (x - 7)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 12 x + 8 = 0$

Step 4

Solve the equation for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 12$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (1 \cdot 8)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 8$ .

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 8}}{2 \cdot 1}$

Multiply $- 4$ by $8$ .

$x = \frac{12 \pm \sqrt{144 - 32}}{2 \cdot 1}$

Subtract $32$ from $144$ .

$x = \frac{12 \pm \sqrt{112}}{2 \cdot 1}$

Rewrite $112$ as $4^{2} \cdot 7$ .

Factor $16$ out of $112$ .

$x = \frac{12 \pm \sqrt{16 (7)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{12 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{12 \pm 4 \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{12 \pm 4 \sqrt{7}}{2}$

Simplify $\frac{12 \pm 4 \sqrt{7}}{2}$ .

$x = 6 \pm 2 \sqrt{7}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 8$ .

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 8}}{2 \cdot 1}$

Multiply $- 4$ by $8$ .

$x = \frac{12 \pm \sqrt{144 - 32}}{2 \cdot 1}$

Subtract $32$ from $144$ .

$x = \frac{12 \pm \sqrt{112}}{2 \cdot 1}$

Rewrite $112$ as $4^{2} \cdot 7$ .

Factor $16$ out of $112$ .

$x = \frac{12 \pm \sqrt{16 (7)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{12 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{12 \pm 4 \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{12 \pm 4 \sqrt{7}}{2}$

Simplify $\frac{12 \pm 4 \sqrt{7}}{2}$ .

$x = 6 \pm 2 \sqrt{7}$

Change the $\pm$ to $+$ .

$x = 6 + 2 \sqrt{7}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 12$ to the power of $2$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 1 \cdot 8}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 8$ .

Multiply $- 4$ by $1$ .

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 8}}{2 \cdot 1}$

Multiply $- 4$ by $8$ .

$x = \frac{12 \pm \sqrt{144 - 32}}{2 \cdot 1}$

Subtract $32$ from $144$ .

$x = \frac{12 \pm \sqrt{112}}{2 \cdot 1}$

Rewrite $112$ as $4^{2} \cdot 7$ .

Factor $16$ out of $112$ .

$x = \frac{12 \pm \sqrt{16 (7)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{12 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{12 \pm 4 \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{12 \pm 4 \sqrt{7}}{2}$

Simplify $\frac{12 \pm 4 \sqrt{7}}{2}$ .

$x = 6 \pm 2 \sqrt{7}$

Change the $\pm$ to $-$ .

$x = 6 - 2 \sqrt{7}$

The final answer is the combination of both solutions.

$x = 6 + 2 \sqrt{7}, 6 - 2 \sqrt{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 6 + 2 \sqrt{7}, 6 - 2 \sqrt{7}$

Decimal Form:

$x = 11.29150262 \dots, 0.70849737 \dots$

Solve by Factoring 3/(x-7)+4/(x+2)=1

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