Solve by Factoring 11/(3x)-1/3=-4/(x^2)

$\frac{11}{3 x} - \frac{1}{3} = \frac{- 4}{x^{2}}$

Step 1

Subtract $\frac{- 4}{x^{2}}$ from both sides of the equation.

$\frac{11}{3 x} - \frac{1}{3} - \frac{- 4}{x^{2}} = 0$

Step 2

Simplify each term.

Move the negative in front of the fraction.

$\frac{11}{3 x} - \frac{1}{3} - - \frac{4}{x^{2}} = 0$

Multiply $- - \frac{4}{x^{2}}$ .

Multiply $- 1$ by $- 1$ .

$\frac{11}{3 x} - \frac{1}{3} + 1 \frac{4}{x^{2}} = 0$

Multiply $\frac{4}{x^{2}}$ by $1$ .

$\frac{11}{3 x} - \frac{1}{3} + \frac{4}{x^{2}} = 0$

Step 3

Reorder terms.

$\frac{11}{3 x} + \frac{4}{x^{2}} - \frac{1}{3} = 0$

Step 4

To write $\frac{11}{3 x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{11}{3 x} \cdot \frac{x}{x} + \frac{4}{x^{2}} - \frac{1}{3} = 0$

Step 5

To write $\frac{4}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{11}{3 x} \cdot \frac{x}{x} + \frac{4}{x^{2}} \cdot \frac{3}{3} - \frac{1}{3} = 0$

Step 6

Write each expression with a common denominator of $3 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{11}{3 x}$ by $\frac{x}{x}$ .

$\frac{11 x}{3 x \cdot x} + \frac{4}{x^{2}} \cdot \frac{3}{3} - \frac{1}{3} = 0$

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

$\frac{11 x}{3 (x \cdot x)} + \frac{4}{x^{2}} \cdot \frac{3}{3} - \frac{1}{3} = 0$

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

$\frac{11 x}{3 (x \cdot x)} + \frac{4}{x^{2}} \cdot \frac{3}{3} - \frac{1}{3} = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{11 x}{3 x^{1 + 1}} + \frac{4}{x^{2}} \cdot \frac{3}{3} - \frac{1}{3} = 0$

Add $1$ and $1$ .

$\frac{11 x}{3 x^{2}} + \frac{4}{x^{2}} \cdot \frac{3}{3} - \frac{1}{3} = 0$

Multiply $\frac{4}{x^{2}}$ by $\frac{3}{3}$ .

$\frac{11 x}{3 x^{2}} + \frac{4 \cdot 3}{x^{2} \cdot 3} - \frac{1}{3} = 0$

Reorder the factors of $x^{2} \cdot 3$ .

$\frac{11 x}{3 x^{2}} + \frac{4 \cdot 3}{3 x^{2}} - \frac{1}{3} = 0$

Step 7

Combine the numerators over the common denominator.

$\frac{11 x + 4 \cdot 3}{3 x^{2}} - \frac{1}{3} = 0$

Step 8

Multiply $4$ by $3$ .

$\frac{11 x + 12}{3 x^{2}} - \frac{1}{3} = 0$

Step 9

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{11 x + 12}{3 x^{2}} - \frac{1}{3} \cdot \frac{x^{2}}{x^{2}} = 0$

Step 10

Multiply $\frac{1}{3}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{11 x + 12}{3 x^{2}} - \frac{x^{2}}{3 x^{2}} = 0$

Step 11

Combine the numerators over the common denominator.

$\frac{11 x + 12 - x^{2}}{3 x^{2}} = 0$

Step 12

Factor by grouping.

Reorder terms.

$\frac{- x^{2} + 11 x + 12}{3 x^{2}} = 0$

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 $a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is $b = 11$ .

Factor $11$ out of $11 x$ .

$\frac{- x^{2} + 11 (x) + 12}{3 x^{2}} = 0$

Rewrite $11$ as $- 1$ plus $12$

$\frac{- x^{2} + (- 1 + 12) x + 12}{3 x^{2}} = 0$

Apply the distributive property.

$\frac{- x^{2} - 1 x + 12 x + 12}{3 x^{2}} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 1 x) + 12 x + 12}{3 x^{2}} = 0$

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

$\frac{x (- x - 1) - 12 (- x - 1)}{3 x^{2}} = 0$

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

$\frac{(- x - 1) (x - 12)}{3 x^{2}} = 0$

Step 13

Set the numerator equal to zero.

$(- x - 1) (x - 12) = 0$

Step 14

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 $0$ .

$- x - 1 = 0$

$x - 12 = 0$

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

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

$- x - 1 = 0$

Solve $- x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$- x = 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $1$ by $- 1$ .

$x = - 1$

Set $x - 12$ equal to $0$ and solve for $x$ .

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Add $12$ to both sides of the equation.

$x = 12$

The final solution is all the values that make $(- x - 1) (x - 12) = 0$ true.

$x = - 1, 12$

Solve by Factoring 11/(3x)-1/3=-4/(x^2)

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