Solve for x (x^2+x)/(x-1)=6

$\frac{x^{2} + x}{x - 1} = 6$

Step 1

Factor $x$ out of $x^{2} + x$ .

Factor $x$ out of $x^{2}$ .

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

Raise $x$ to the power of $1$ .

$\frac{x \cdot x + x^{1}}{x - 1} = 6$

Factor $x$ out of $x^{1}$ .

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

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$\frac{x (x + 1)}{x - 1} = 6$

Step 2

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.

$x - 1, 1$

Remove parentheses.

$x - 1, 1$

The LCM of one and any expression is the expression.

$x - 1$

Step 3

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

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

$\frac{x (x + 1)}{x - 1} (x - 1) = 6 (x - 1)$

Simplify the left side.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

$\frac{x (x + 1)}{x - 1} (x - 1) = 6 (x - 1)$

Rewrite the expression.

$x (x + 1) = 6 (x - 1)$

Apply the distributive property.

$x \cdot x + x \cdot 1 = 6 (x - 1)$

Simplify the expression.

Multiply $x$ by $x$ .

$x^{2} + x \cdot 1 = 6 (x - 1)$

Multiply $x$ by $1$ .

$x^{2} + x = 6 (x - 1)$

Simplify the right side.

Apply the distributive property.

$x^{2} + x = 6 x + 6 \cdot - 1$

Multiply $6$ by $- 1$ .

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

Step 4

Solve the equation.

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

Subtract $6 x$ from both sides of the equation.

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

Subtract $6 x$ from $x$ .

$x^{2} - 5 x = - 6$

Add $6$ to both sides of the equation.

$x^{2} - 5 x + 6 = 0$

Factor $x^{2} - 5 x + 6$ 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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

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

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

$x - 3 = 0$

$x - 2 = 0$

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

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

$x = 3, 2$

Solve for x (x^2+x)/(x-1)=6

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