Solve by Factoring 5/(x^2-4x+3)=(2x)/(x-1)-x/(x-3)

$\frac{5}{x^{2} - 4 x + 3} = \frac{2 x}{x - 1} - \frac{x}{x - 3}$

Step 1

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

Subtract $\frac{2 x}{x - 1}$ from both sides of the equation.

$\frac{5}{x^{2} - 4 x + 3} - \frac{2 x}{x - 1} = - \frac{x}{x - 3}$

Add $\frac{x}{x - 3}$ to both sides of the equation.

$\frac{5}{x^{2} - 4 x + 3} - \frac{2 x}{x - 1} + \frac{x}{x - 3} = 0$

Step 2

Simplify $\frac{5}{x^{2} - 4 x + 3} - \frac{2 x}{x - 1} + \frac{x}{x - 3}$ .

Factor $x^{2} - 4 x + 3$ 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 $3$ and whose sum is $- 4$ .

$- 3, - 1$

Write the factored form using these integers.

$\frac{5}{(x - 3) (x - 1)} - \frac{2 x}{x - 1} + \frac{x}{x - 3} = 0$

Find the common denominator.

Multiply $\frac{2 x}{x - 1}$ by $\frac{x - 3}{x - 3}$ .

$\frac{5}{(x - 3) (x - 1)} - (\frac{2 x}{x - 1} \cdot \frac{x - 3}{x - 3}) + \frac{x}{x - 3} = 0$

Multiply $\frac{2 x}{x - 1}$ by $\frac{x - 3}{x - 3}$ .

$\frac{5}{(x - 3) (x - 1)} - \frac{2 x (x - 3)}{(x - 1) (x - 3)} + \frac{x}{x - 3} = 0$

Multiply $\frac{x}{x - 3}$ by $\frac{x - 1}{x - 1}$ .

$\frac{5}{(x - 3) (x - 1)} - \frac{2 x (x - 3)}{(x - 1) (x - 3)} + \frac{x}{x - 3} \cdot \frac{x - 1}{x - 1} = 0$

Multiply $\frac{x}{x - 3}$ by $\frac{x - 1}{x - 1}$ .

$\frac{5}{(x - 3) (x - 1)} - \frac{2 x (x - 3)}{(x - 1) (x - 3)} + \frac{x (x - 1)}{(x - 3) (x - 1)} = 0$

Combine the numerators over the common denominator.

$\frac{5 - 2 x (x - 3) + x (x - 1)}{(x - 1) (x - 3)} = 0$

Simplify each term.

Apply the distributive property.

$\frac{5 - 2 x \cdot x - 2 x \cdot - 3 + x (x - 1)}{(x - 1) (x - 3)} = 0$

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

Move $x$ .

$\frac{5 - 2 (x \cdot x) - 2 x \cdot - 3 + x (x - 1)}{(x - 1) (x - 3)} = 0$

Multiply $x$ by $x$ .

$\frac{5 - 2 x^{2} - 2 x \cdot - 3 + x (x - 1)}{(x - 1) (x - 3)} = 0$

Multiply $- 3$ by $- 2$ .

$\frac{5 - 2 x^{2} + 6 x + x (x - 1)}{(x - 1) (x - 3)} = 0$

Apply the distributive property.

$\frac{5 - 2 x^{2} + 6 x + x \cdot x + x \cdot - 1}{(x - 1) (x - 3)} = 0$

Multiply $x$ by $x$ .

$\frac{5 - 2 x^{2} + 6 x + x^{2} + x \cdot - 1}{(x - 1) (x - 3)} = 0$

Move $- 1$ to the left of $x$ .

$\frac{5 - 2 x^{2} + 6 x + x^{2} - 1 \cdot x}{(x - 1) (x - 3)} = 0$

Rewrite $- 1 x$ as $- x$ .

$\frac{5 - 2 x^{2} + 6 x + x^{2} - x}{(x - 1) (x - 3)} = 0$

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

$\frac{5 - x^{2} + 6 x - x}{(x - 1) (x - 3)} = 0$

Subtract $x$ from $6 x$ .

$\frac{5 - x^{2} + 5 x}{(x - 1) (x - 3)} = 0$

Reorder terms.

$\frac{- x^{2} + 5 x + 5}{(x - 1) (x - 3)} = 0$

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

$\frac{- (x^{2}) + 5 x + 5}{(x - 1) (x - 3)} = 0$

Factor $- 1$ out of $5 x$ .

$\frac{- (x^{2}) - (- 5 x) + 5}{(x - 1) (x - 3)} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 5 x)$ .

$\frac{- (x^{2} - 5 x) + 5}{(x - 1) (x - 3)} = 0$

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{- (x^{2} - 5 x) - 1 (- 5)}{(x - 1) (x - 3)} = 0$

Factor $- 1$ out of $- (x^{2} - 5 x) - 1 (- 5)$ .

$\frac{- (x^{2} - 5 x - 5)}{(x - 1) (x - 3)} = 0$

Rewrite $- (x^{2} - 5 x - 5)$ as $- 1 (x^{2} - 5 x - 5)$ .

$\frac{- 1 (x^{2} - 5 x - 5)}{(x - 1) (x - 3)} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 5 x - 5}{(x - 1) (x - 3)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 5 x - 5 = 0$

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 $a = 1$ , $b = - 5$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4$ by $- 5$ .

$x = \frac{5 \pm \sqrt{25 + 20}}{2 \cdot 1}$

Add $25$ and $20$ .

$x = \frac{5 \pm \sqrt{45}}{2 \cdot 1}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$x = \frac{5 \pm \sqrt{9 (5)}}{2 \cdot 1}$

Rewrite $9$ as $3^{2}$ .

$x = \frac{5 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{5 \pm 3 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{5 \pm 3 \sqrt{5}}{2}$

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

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4$ by $- 5$ .

$x = \frac{5 \pm \sqrt{25 + 20}}{2 \cdot 1}$

Add $25$ and $20$ .

$x = \frac{5 \pm \sqrt{45}}{2 \cdot 1}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$x = \frac{5 \pm \sqrt{9 (5)}}{2 \cdot 1}$

Rewrite $9$ as $3^{2}$ .

$x = \frac{5 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{5 \pm 3 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{5 \pm 3 \sqrt{5}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{5 + 3 \sqrt{5}}{2}$

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

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4$ by $- 5$ .

$x = \frac{5 \pm \sqrt{25 + 20}}{2 \cdot 1}$

Add $25$ and $20$ .

$x = \frac{5 \pm \sqrt{45}}{2 \cdot 1}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$x = \frac{5 \pm \sqrt{9 (5)}}{2 \cdot 1}$

Rewrite $9$ as $3^{2}$ .

$x = \frac{5 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{5 \pm 3 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{5 \pm 3 \sqrt{5}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{5 - 3 \sqrt{5}}{2}$

The final answer is the combination of both solutions.

$x = \frac{5 + 3 \sqrt{5}}{2}, \frac{5 - 3 \sqrt{5}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + 3 \sqrt{5}}{2}, \frac{5 - 3 \sqrt{5}}{2}$

Decimal Form:

$x = 5.85410196 \dots, - 0.85410196 \dots$

Solve by Factoring 5/(x^2-4x+3)=(2x)/(x-1)-x/(x-3)

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