Solve by Factoring 1/x-6/(x^2)=-12

$\frac{1}{x} - \frac{6}{x^{2}} = - 12$

Step 1

Add $12$ to both sides of the equation.

$\frac{1}{x} - \frac{6}{x^{2}} + 12 = 0$

Step 2

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

$\frac{1}{x} \cdot \frac{x}{x} - \frac{6}{x^{2}} + 12 = 0$

Step 3

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

Multiply $\frac{1}{x}$ by $\frac{x}{x}$ .

$\frac{x}{x \cdot x} - \frac{6}{x^{2}} + 12 = 0$

Multiply $x$ by $x$ .

$\frac{x}{x^{2}} - \frac{6}{x^{2}} + 12 = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{x - 6}{x^{2}} + 12 = 0$

Step 5

To write $12$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{x - 6}{x^{2}} + \frac{12 x^{2}}{x^{2}} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{x - 6 + 12 x^{2}}{x^{2}} = 0$

Step 7

Factor by grouping.

Reorder terms.

$\frac{12 x^{2} + x - 6}{x^{2}} = 0$

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 $a \cdot c = 12 \cdot - 6 = - 72$ and whose sum is $b = 1$ .

Multiply by $1$ .

$\frac{12 x^{2} + 1 x - 6}{x^{2}} = 0$

Rewrite $1$ as $- 8$ plus $9$

$\frac{12 x^{2} + (- 8 + 9) x - 6}{x^{2}} = 0$

Apply the distributive property.

$\frac{12 x^{2} - 8 x + 9 x - 6}{x^{2}} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(12 x^{2} - 8 x) + 9 x - 6}{x^{2}} = 0$

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

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

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

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

Step 8

Set the numerator equal to zero.

$(3 x - 2) (4 x + 3) = 0$

Step 9

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 $0$ .

$3 x - 2 = 0$

$4 x + 3 = 0$

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 $4 x + 3$ equal to $0$ and solve for $x$ .

Set $4 x + 3$ equal to $0$ .

$4 x + 3 = 0$

Solve $4 x + 3 = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$4 x = - 3$

Divide each term in $4 x = - 3$ by $4$ and simplify.

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

The final solution is all the values that make $(3 x - 2) (4 x + 3) = 0$ true.

$x = \frac{2}{3}, - \frac{3}{4}$

Solve by Factoring 1/x-6/(x^2)=-12

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