Algebra Examples

$x - 9 = - \frac{18}{x}$

Step 1

Add $\frac{18}{x}$ to both sides of the equation.

$x - 9 + \frac{18}{x} = 0$

Step 2

Reorder terms.

$x + \frac{18}{x} - 9 = 0$

Step 3

To write $x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x \cdot x}{x} + \frac{18}{x} - 9 = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{x \cdot x + 18}{x} - 9 = 0$

Step 5

Multiply $x$ by $x$ .

$\frac{x^{2} + 18}{x} - 9 = 0$

Step 6

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x^{2} + 18}{x} - 9 \cdot \frac{x}{x} = 0$

Step 7

Combine $- 9$ and $\frac{x}{x}$ .

$\frac{x^{2} + 18}{x} + \frac{- 9 x}{x} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{x^{2} + 18 - 9 x}{x} = 0$

Step 9

Factor $x^{2} + 18 - 9 x$ using the AC method.

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Step 9.1

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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Step 9.2

Write the factored form using these integers.

$\frac{(x - 6) (x - 3)}{x} = 0$

Step 10

Set the numerator equal to zero.

$(x - 6) (x - 3) = 0$

Step 11

Solve the equation for $x$ .

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Step 11.1

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

$x - 6 = 0$

$x - 3 = 0$

Step 11.2

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 11.2.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 11.2.2

Add $6$ to both sides of the equation.

$x = 6$

Step 11.3

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

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Step 11.3.1

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

$x - 3 = 0$

Step 11.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 11.4

The final solution is all the values that make $(x - 6) (x - 3) = 0$ true.

$x = 6, 3$