Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Factor $x^{2} - 4 x + 3$ using the AC method.

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Step 2.1.1

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 $- 4$ .

$- 3, - 1$

Step 2.1.2

Write the factored form using these integers.

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

Step 2.2

Find the common denominator.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Simplify each term.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

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

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Step 2.4.2.1

Move $x$ .

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

Step 2.4.2.2

Multiply $x$ by $x$ .

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

Step 2.4.3

Multiply $- 3$ by $- 2$ .

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

Step 2.4.4

Apply the distributive property.

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

Step 2.4.5

Multiply $x$ by $x$ .

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

Step 2.4.6

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

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

Step 2.4.7

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

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

Step 2.5

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

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

Step 2.6

Subtract $x$ from $6 x$ .

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

Step 2.7

Reorder terms.

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 2.12

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

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

Step 2.13

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

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

Step 2.14

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

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

Step 4

Solve the equation for $x$ .

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Step 4.1

Use the quadratic formula to find the solutions.

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

Step 4.2

Substitute the values $a = 1$ , $b = - 5$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

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

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

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

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

Step 4.3.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 4.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 4.3.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{5 \pm \sqrt{25 + 8}}{2 \cdot 1}$

Step 4.3.1.3

Add $25$ and $8$ .

$x = \frac{5 \pm \sqrt{33}}{2 \cdot 1}$

Step 4.3.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{33}}{2}$

Step 4.4

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

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

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

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

Step 4.4.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 4.4.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{5 \pm \sqrt{25 + 8}}{2 \cdot 1}$

Step 4.4.1.3

Add $25$ and $8$ .

$x = \frac{5 \pm \sqrt{33}}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{33}}{2}$

Step 4.4.3

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{33}}{2}$

Step 4.5

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

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

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

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

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot - 2$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 4.5.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{5 \pm \sqrt{25 + 8}}{2 \cdot 1}$

Step 4.5.1.3

Add $25$ and $8$ .

$x = \frac{5 \pm \sqrt{33}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{33}}{2}$

Step 4.5.3

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{33}}{2}$

Step 4.6

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{33}}{2}, \frac{5 - \sqrt{33}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{33}}{2}, \frac{5 - \sqrt{33}}{2}$

Decimal Form:

$x = 5.37228132 \dots, - 0.37228132 \dots$