Find All Complex Solutions x+4=(x-2)/x

$x + 4 = \frac{x - 2}{x}$

Step 1

Multiply each term by a factor of $1$ that will equate all the denominators. In this case, all terms need a denominator of $x$ .

$x \cdot \frac{x}{x} + 4 \cdot \frac{x}{x} = \frac{x - 2}{x}$

Step 2

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $x$ .

$x \cdot x$

Step 3

Multiply $x$ by $x$ .

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

Step 4

Multiply the expression by a factor of $1$ to create the least common denominator (LCD) of $x$ .

$4 \cdot x$

Step 5

Multiply $4$ by $x$ .

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

Step 6

Factor each term.

Multiply $x$ by $\frac{x}{x}$ .

$x \frac{x}{x} + 4 \cdot \frac{x}{x} = \frac{x - 2}{x}$

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

$\frac{x \cdot x}{x} + 4 \cdot \frac{x}{x} = \frac{x - 2}{x}$

Multiply $4$ by $\frac{x}{x}$ .

$\frac{x \cdot x}{x} + 4 \frac{x}{x} = \frac{x - 2}{x}$

Combine $4$ and $\frac{x}{x}$ .

$\frac{x \cdot x}{x} + \frac{4 x}{x} = \frac{x - 2}{x}$

Reduce the expression $\frac{x \cdot x}{x}$ by cancelling the common factors.

Cancel the common factor.

$\frac{x \cdot x}{x} + \frac{4 x}{x} = \frac{x - 2}{x}$

Rewrite the expression.

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

Reduce the expression $\frac{4 x}{x}$ by cancelling the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Divide $x$ by $1$ .

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

Divide $4$ by $1$ .

$x + 4 = \frac{x - 2}{x}$

Step 7

Find the LCD of the terms in the equation.

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

$1, 1, x$

The LCM of one and any expression is the expression.

$x$

Step 8

Multiply each term in $x + 4 = \frac{x - 2}{x}$ by $x$ to eliminate the fractions.

Multiply each term in $x + 4 = \frac{x - 2}{x}$ by $x$ .

$x \cdot x + 4 x = \frac{x - 2}{x} x$

Simplify the left side.

Multiply $x$ by $x$ .

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

Simplify the right side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 9

Solve the equation.

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

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $4 x$ .

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

Add $2$ to both sides of the equation.

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

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

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 $2$ and whose sum is $3$ .

$1, 2$

Write the factored form using these integers.

$(x + 1) (x + 2) = 0$

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

$x + 1 = 0$

$x + 2 = 0$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $(x + 1) (x + 2) = 0$ true.

$x = - 1, - 2$

Find All Complex Solutions x+4=(x-2)/x

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