Algebra Examples

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

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 1, x - 1, 1$

Step 1.2

Remove parentheses.

$1, 1, x - 1, 1$

Step 1.3

The LCM of one and any expression is the expression.

$x - 1$

Step 2

Multiply each term in $x + 3 - \frac{4}{x - 1} = 5$ by $x - 1$ to eliminate the fractions.

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Step 2.1

Multiply each term in $x + 3 - \frac{4}{x - 1} = 5$ by $x - 1$ .

$x (x - 1) + 3 (x - 1) - \frac{4}{x - 1} (x - 1) = 5 (x - 1)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Apply the distributive property.

$x \cdot x + x \cdot - 1 + 3 (x - 1) - \frac{4}{x - 1} (x - 1) = 5 (x - 1)$

Step 2.2.1.2

Multiply $x$ by $x$ .

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

Step 2.2.1.3

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

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

Step 2.2.1.4

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

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

Step 2.2.1.5

Apply the distributive property.

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

Step 2.2.1.6

Multiply $3$ by $- 1$ .

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

Step 2.2.1.7

Cancel the common factor of $x - 1$ .

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Step 2.2.1.7.1

Move the leading negative in $- \frac{4}{x - 1}$ into the numerator.

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

Step 2.2.1.7.2

Cancel the common factor.

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

Step 2.2.1.7.3

Rewrite the expression.

$x^{2} - x + 3 x - 3 - 4 = 5 (x - 1)$

Step 2.2.2

Simplify by adding terms.

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Step 2.2.2.1

Add $- x$ and $3 x$ .

$x^{2} + 2 x - 3 - 4 = 5 (x - 1)$

Step 2.2.2.2

Subtract $4$ from $- 3$ .

$x^{2} + 2 x - 7 = 5 (x - 1)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Apply the distributive property.

$x^{2} + 2 x - 7 = 5 x + 5 \cdot - 1$

Step 2.3.2

Multiply $5$ by $- 1$ .

$x^{2} + 2 x - 7 = 5 x - 5$

Step 3

Solve the equation.

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $5 x$ from both sides of the equation.

$x^{2} + 2 x - 7 - 5 x = - 5$

Step 3.1.2

Subtract $5 x$ from $2 x$ .

$x^{2} - 3 x - 7 = - 5$

Step 3.2

Move all terms to the left side of the equation and simplify.

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Step 3.2.1

Add $5$ to both sides of the equation.

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

Step 3.2.2

Add $- 7$ and $5$ .

$x^{2} - 3 x - 2 = 0$

Step 3.3

Use the quadratic formula to find the solutions.

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

Step 3.4

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 2)}}{2 \cdot 1}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 2}}{2 \cdot 1}$

Step 3.5.1.2

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

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Step 3.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 2}}{2 \cdot 1}$

Step 3.5.1.2.2

Multiply $- 4$ by $- 2$ .

$x = \frac{3 \pm \sqrt{9 + 8}}{2 \cdot 1}$

Step 3.5.1.3

Add $9$ and $8$ .

$x = \frac{3 \pm \sqrt{17}}{2 \cdot 1}$

Step 3.5.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{17}}{2}$

Step 3.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{17}}{2}, \frac{3 - \sqrt{17}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{17}}{2}, \frac{3 - \sqrt{17}}{2}$

Decimal Form:

$x = 3.56155281 \dots, - 0.56155281 \dots$