Algebra Examples

$x - 4 = \frac{5}{x}$

Step 1

Add $4$ to both sides of the equation.

$x = \frac{5}{x} + 4$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$1, x, 1$

Step 2.2

The LCM of one and any expression is the expression.

$x$

Step 3

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

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Step 3.1

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

$x \cdot x = \frac{5}{x} x + 4 x$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply $x$ by $x$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $x$ .

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Step 3.3.1.1

Cancel the common factor.

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

Step 3.3.1.2

Rewrite the expression.

$x^{2} = 5 + 4 x$

Step 4

Solve the equation.

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Step 4.1

Subtract $4 x$ from both sides of the equation.

$x^{2} - 4 x = 5$

Step 4.2

Subtract $5$ from both sides of the equation.

$x^{2} - 4 x - 5 = 0$

Step 4.3

Factor $x^{2} - 4 x - 5$ using the AC method.

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Step 4.3.1

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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 4.3.2

Write the factored form using these integers.

$(x - 5) (x + 1) = 0$

Step 4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 5 = 0$

$x + 1 = 0$

Step 4.5

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.5.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.6

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 4.6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4.7

The final solution is all the values that make $(x - 5) (x + 1) = 0$ true.

$x = 5, - 1$