Solve by Factoring x+1/x=13/6

$x + \frac{1}{x} = \frac{13}{6}$

Step 1

Subtract $\frac{13}{6}$ from both sides of the equation.

$x + \frac{1}{x} - \frac{13}{6} = 0$

Step 2

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

$\frac{x \cdot x}{x} + \frac{1}{x} - \frac{13}{6} = 0$

Step 3

Combine the numerators over the common denominator.

$\frac{x \cdot x + 1}{x} - \frac{13}{6} = 0$

Step 4

Multiply $x$ by $x$ .

$\frac{x^{2} + 1}{x} - \frac{13}{6} = 0$

Step 5

To write $\frac{x^{2} + 1}{x}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{x^{2} + 1}{x} \cdot \frac{6}{6} - \frac{13}{6} = 0$

Step 6

To write $- \frac{13}{6}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x^{2} + 1}{x} \cdot \frac{6}{6} - \frac{13}{6} \cdot \frac{x}{x} = 0$

Step 7

Write each expression with a common denominator of $x \cdot 6$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x^{2} + 1}{x}$ by $\frac{6}{6}$ .

$\frac{(x^{2} + 1) \cdot 6}{x \cdot 6} - \frac{13}{6} \cdot \frac{x}{x} = 0$

Multiply $\frac{13}{6}$ by $\frac{x}{x}$ .

$\frac{(x^{2} + 1) \cdot 6}{x \cdot 6} - \frac{13 x}{6 x} = 0$

Reorder the factors of $x \cdot 6$ .

$\frac{(x^{2} + 1) \cdot 6}{6 x} - \frac{13 x}{6 x} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{(x^{2} + 1) \cdot 6 - 13 x}{6 x} = 0$

Step 9

Simplify the numerator.

Apply the distributive property.

$\frac{x^{2} \cdot 6 + 1 \cdot 6 - 13 x}{6 x} = 0$

Move $6$ to the left of $x^{2}$ .

$\frac{6 \cdot x^{2} + 1 \cdot 6 - 13 x}{6 x} = 0$

Multiply $6$ by $1$ .

$\frac{6 x^{2} + 6 - 13 x}{6 x} = 0$

Reorder terms.

$\frac{6 x^{2} - 13 x + 6}{6 x} = 0$

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 $a \cdot c = 6 \cdot 6 = 36$ and whose sum is $b = - 13$ .

Factor $- 13$ out of $- 13 x$ .

$\frac{6 x^{2} - 13 x + 6}{6 x} = 0$

Rewrite $- 13$ as $- 4$ plus $- 9$

$\frac{6 x^{2} + (- 4 - 9) x + 6}{6 x} = 0$

Apply the distributive property.

$\frac{6 x^{2} - 4 x - 9 x + 6}{6 x} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(6 x^{2} - 4 x) - 9 x + 6}{6 x} = 0$

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

$\frac{2 x (3 x - 2) - 3 (3 x - 2)}{6 x} = 0$

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

$\frac{(3 x - 2) (2 x - 3)}{6 x} = 0$

Step 10

Set the numerator equal to zero.

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

Step 11

Solve the equation for $x$ .

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

$3 x - 2 = 0$

$2 x - 3 = 0$

Set $3 x - 2$ equal to $0$ and solve for $x$ .

Set $3 x - 2$ equal to $0$ .

$3 x - 2 = 0$

Solve $3 x - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

$3 x = 2$

Divide each term in $3 x = 2$ by $3$ and simplify.

Divide each term in $3 x = 2$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Set $2 x - 3$ equal to $0$ and solve for $x$ .

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Solve $2 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

Divide each term in $2 x = 3$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

$x = \frac{2}{3}, \frac{3}{2}$

Solve by Factoring x+1/x=13/6

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