Solve the Rational Equation for x (5x^2+2x+5)/(x+2)=5x+1

$\frac{5 x^{2} + 2 x + 5}{x + 2} = 5 x + 1$

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.

$x + 2, 1, 1$

Remove parentheses.

$x + 2, 1, 1$

The LCM of one and any expression is the expression.

$x + 2$

Multiply each term in $\frac{5 x^{2} + 2 x + 5}{x + 2} = 5 x + 1$ by $x + 2$ to eliminate the fractions.

Multiply each term in $\frac{5 x^{2} + 2 x + 5}{x + 2} = 5 x + 1$ by $x + 2$ .

$\frac{5 x^{2} + 2 x + 5}{x + 2} (x + 2) = 5 x (x + 2) + 1 (x + 2)$

Simplify the left side.

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{5 x^{2} + 2 x + 5}{x + 2} (x + 2) = 5 x (x + 2) + 1 (x + 2)$

Rewrite the expression.

$5 x^{2} + 2 x + 5 = 5 x (x + 2) + 1 (x + 2)$

Simplify the right side.

Simplify each term.

Apply the distributive property.

$5 x^{2} + 2 x + 5 = 5 x \cdot x + 5 x \cdot 2 + 1 (x + 2)$

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

Move $x$ .

$5 x^{2} + 2 x + 5 = 5 (x \cdot x) + 5 x \cdot 2 + 1 (x + 2)$

Multiply $x$ by $x$ .

$5 x^{2} + 2 x + 5 = 5 x^{2} + 5 x \cdot 2 + 1 (x + 2)$

Multiply $2$ by $5$ .

$5 x^{2} + 2 x + 5 = 5 x^{2} + 10 x + 1 (x + 2)$

Multiply $x + 2$ by $1$ .

$5 x^{2} + 2 x + 5 = 5 x^{2} + 10 x + x + 2$

Add $10 x$ and $x$ .

$5 x^{2} + 2 x + 5 = 5 x^{2} + 11 x + 2$

Solve the equation.

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

Subtract $5 x^{2}$ from both sides of the equation.

$5 x^{2} + 2 x + 5 - 5 x^{2} = 11 x + 2$

Subtract $11 x$ from both sides of the equation.

$5 x^{2} + 2 x + 5 - 5 x^{2} - 11 x = 2$

Combine the opposite terms in $5 x^{2} + 2 x + 5 - 5 x^{2} - 11 x$ .

Subtract $5 x^{2}$ from $5 x^{2}$ .

$2 x + 5 + 0 - 11 x = 2$

Add $2 x + 5$ and $0$ .

$2 x + 5 - 11 x = 2$

Subtract $11 x$ from $2 x$ .

$- 9 x + 5 = 2$

Move all terms not containing $x$ to the right side of the equation.

Subtract $5$ from both sides of the equation.

$- 9 x = 2 - 5$

Subtract $5$ from $2$ .

$- 9 x = - 3$

Divide each term in $- 9 x = - 3$ by $- 9$ and simplify.

Divide each term in $- 9 x = - 3$ by $- 9$ .

$\frac{- 9 x}{- 9} = \frac{- 3}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 x}{- 9} = \frac{- 3}{- 9}$

Divide $x$ by $1$ .

$x = \frac{- 3}{- 9}$

Simplify the right side.

Cancel the common factor of $- 3$ and $- 9$ .

Factor $- 3$ out of $- 3$ .

$x = \frac{- 3 (1)}{- 9}$

Cancel the common factors.

Factor $- 3$ out of $- 9$ .

$x = \frac{- 3 \cdot 1}{- 3 \cdot 3}$

Cancel the common factor.

$x = \frac{- 3 \cdot 1}{- 3 \cdot 3}$

Rewrite the expression.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{3}$

Solve the Rational Equation for x (5x^2+2x+5)/(x+2)=5x+1

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