Solve for x 2x+3/(x+2)=1

$2 x + \frac{3}{x + 2} = 1$

Step 1

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.

$1, x + 2, 1$

Remove parentheses.

$1, x + 2, 1$

The LCM of one and any expression is the expression.

$x + 2$

Step 2

Multiply each term in $2 x + \frac{3}{x + 2} = 1$ by $x + 2$ to eliminate the fractions.

Multiply each term in $2 x + \frac{3}{x + 2} = 1$ by $x + 2$ .

$2 x (x + 2) + \frac{3}{x + 2} (x + 2) = 1 (x + 2)$

Simplify the left side.

Simplify each term.

Apply the distributive property.

$2 x \cdot x + 2 x \cdot 2 + \frac{3}{x + 2} (x + 2) = 1 (x + 2)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$2 (x \cdot x) + 2 x \cdot 2 + \frac{3}{x + 2} (x + 2) = 1 (x + 2)$

Multiply $x$ by $x$ .

$2 x^{2} + 2 x \cdot 2 + \frac{3}{x + 2} (x + 2) = 1 (x + 2)$

Multiply $2$ by $2$ .

$2 x^{2} + 4 x + \frac{3}{x + 2} (x + 2) = 1 (x + 2)$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$2 x^{2} + 4 x + \frac{3}{x + 2} (x + 2) = 1 (x + 2)$

Rewrite the expression.

$2 x^{2} + 4 x + 3 = 1 (x + 2)$

Simplify the right side.

Multiply $x + 2$ by $1$ .

$2 x^{2} + 4 x + 3 = x + 2$

Step 3

Solve the equation.

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

Subtract $x$ from both sides of the equation.

$2 x^{2} + 4 x + 3 - x = 2$

Subtract $x$ from $4 x$ .

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

Subtract $2$ from both sides of the equation.

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

Subtract $2$ from $3$ .

$2 x^{2} + 3 x + 1 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 1 = 2$ and whose sum is $b = 3$ .

Factor $3$ out of $3 x$ .

$2 x^{2} + 3 (x) + 1 = 0$

Rewrite $3$ as $1$ plus $2$

$2 x^{2} + (1 + 2) x + 1 = 0$

Apply the distributive property.

$2 x^{2} + 1 x + 2 x + 1 = 0$

Multiply $x$ by $1$ .

$2 x^{2} + x + 2 x + 1 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 x^{2} + x) + 2 x + 1 = 0$

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

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

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

$2 x + 1 = 0$

$x + 1 = 0$

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve $2 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term in $2 x = - 1$ by $2$ and simplify.

Divide each term in $2 x = - 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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.

$x = - 1$

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

$x = - \frac{1}{2}, - 1$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}, - 1$

Decimal Form:

$x = - 0.5, - 1$

Solve for x 2x+3/(x+2)=1

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