Solve by Factoring 7/(x^2-5x+6)=(2x)/(x-3)-x/(x-2)

$\frac{7}{x^{2} - 5 x + 6} = \frac{2 x}{x - 3} - \frac{x}{x - 2}$

Step 1

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

Subtract $\frac{2 x}{x - 3}$ from both sides of the equation.

$\frac{7}{x^{2} - 5 x + 6} - \frac{2 x}{x - 3} = - \frac{x}{x - 2}$

Add $\frac{x}{x - 2}$ to both sides of the equation.

$\frac{7}{x^{2} - 5 x + 6} - \frac{2 x}{x - 3} + \frac{x}{x - 2} = 0$

Step 2

Simplify $\frac{7}{x^{2} - 5 x + 6} - \frac{2 x}{x - 3} + \frac{x}{x - 2}$ .

Factor $x^{2} - 5 x + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

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

Find the common denominator.

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 2$ .

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

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

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

Add $- 2 x^{2}$ and $x^{2}$ .

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

Subtract $3 x$ from $4 x$ .

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

Reorder terms.

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

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

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

Factor $- 1$ out of $x$ .

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

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

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

Rewrite $7$ as $- 1 (- 7)$ .

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

Factor $- 1$ out of $- (x^{2} - x) - 1 (- 7)$ .

$\frac{- (x^{2} - x - 7)}{(x - 2) (x - 3)} = 0$

Rewrite $- (x^{2} - x - 7)$ as $- 1 (x^{2} - x - 7)$ .

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

Move the negative in front of the fraction.

$- \frac{x^{2} - x - 7}{(x - 2) (x - 3)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - x - 7 = 0$

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 $a = 1$ , $b = - 1$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot - 7)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

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

Multiply $- 4 \cdot 1 \cdot - 7$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 7$ .

$x = \frac{1 \pm \sqrt{1 + 28}}{2 \cdot 1}$

Add $1$ and $28$ .

$x = \frac{1 \pm \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{29}}{2}$

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

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

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

Multiply $- 4 \cdot 1 \cdot - 7$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 7$ .

$x = \frac{1 \pm \sqrt{1 + 28}}{2 \cdot 1}$

Add $1$ and $28$ .

$x = \frac{1 \pm \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{29}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{29}}{2}$

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

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

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

Multiply $- 4 \cdot 1 \cdot - 7$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 7$ .

$x = \frac{1 \pm \sqrt{1 + 28}}{2 \cdot 1}$

Add $1$ and $28$ .

$x = \frac{1 \pm \sqrt{29}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{1 \pm \sqrt{29}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{29}}{2}$

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{29}}{2}, \frac{1 - \sqrt{29}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{29}}{2}, \frac{1 - \sqrt{29}}{2}$

Decimal Form:

$x = 3.19258240 \dots, - 2.19258240 \dots$

Solve by Factoring 7/(x^2-5x+6)=(2x)/(x-3)-x/(x-2)

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