Precalculus Examples

$x - \frac{6}{x} = - 1$

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.

$1, x, 1$

Step 1.2

The LCM of one and any expression is the expression.

$x$

Step 2

Multiply each term in $x - \frac{6}{x} = - 1$ by $x$ to eliminate the fractions.

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Step 2.1

Multiply each term in $x - \frac{6}{x} = - 1$ by $x$ .

$x \cdot x - \frac{6}{x} x = - x$

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

Multiply $x$ by $x$ .

$x^{2} - \frac{6}{x} x = - x$

Step 2.2.1.2

Cancel the common factor of $x$ .

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Step 2.2.1.2.1

Move the leading negative in $- \frac{6}{x}$ into the numerator.

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

Step 2.2.1.2.2

Cancel the common factor.

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

Step 2.2.1.2.3

Rewrite the expression.

$x^{2} - 6 = - x$

Step 3

Solve the equation.

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Step 3.1

Add $x$ to both sides of the equation.

$x^{2} - 6 + x = 0$

Step 3.2

Factor $x^{2} - 6 + x$ using the AC method.

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Step 3.2.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 3.2.2

Write the factored form using these integers.

$(x - 2) (x + 3) = 0$

Step 3.3

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

$x - 2 = 0$

$x + 3 = 0$

Step 3.4

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 3.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3.6

The final solution is all the values that make $(x - 2) (x + 3) = 0$ true.

$x = 2, - 3$