Solve by Factoring 6/(x^2)-49/x=17

$\frac{6}{x^{2}} - \frac{49}{x} = 17$

Step 1

Subtract $17$ from both sides of the equation.

$\frac{6}{x^{2}} - \frac{49}{x} - 17 = 0$

Step 2

Reorder terms.

$- \frac{49}{x} + \frac{6}{x^{2}} - 17 = 0$

Step 3

To write $- \frac{49}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{49}{x} \cdot \frac{x}{x} + \frac{6}{x^{2}} - 17 = 0$

Step 4

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

Multiply $\frac{49}{x}$ by $\frac{x}{x}$ .

$- \frac{49 x}{x \cdot x} + \frac{6}{x^{2}} - 17 = 0$

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

$- \frac{49 x}{x \cdot x} + \frac{6}{x^{2}} - 17 = 0$

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

$- \frac{49 x}{x \cdot x} + \frac{6}{x^{2}} - 17 = 0$

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

$- \frac{49 x}{x^{1 + 1}} + \frac{6}{x^{2}} - 17 = 0$

Add $1$ and $1$ .

$- \frac{49 x}{x^{2}} + \frac{6}{x^{2}} - 17 = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{- 49 x + 6}{x^{2}} - 17 = 0$

Step 6

To write $- 17$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{- 49 x + 6}{x^{2}} - 17 \cdot \frac{x^{2}}{x^{2}} = 0$

Step 7

Combine $- 17$ and $\frac{x^{2}}{x^{2}}$ .

$\frac{- 49 x + 6}{x^{2}} + \frac{- 17 x^{2}}{x^{2}} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{- 49 x + 6 - 17 x^{2}}{x^{2}} = 0$

Step 9

Factor by grouping.

Reorder terms.

$\frac{- 17 x^{2} - 49 x + 6}{x^{2}} = 0$

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 $a \cdot c = - 17 \cdot 6 = - 102$ and whose sum is $b = - 49$ .

Factor $- 49$ out of $- 49 x$ .

$\frac{- 17 x^{2} - 49 x + 6}{x^{2}} = 0$

Rewrite $- 49$ as $2$ plus $- 51$

$\frac{- 17 x^{2} + (2 - 51) x + 6}{x^{2}} = 0$

Apply the distributive property.

$\frac{- 17 x^{2} + 2 x - 51 x + 6}{x^{2}} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- 17 x^{2} + 2 x) - 51 x + 6}{x^{2}} = 0$

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

$\frac{x (- 17 x + 2) + 3 (- 17 x + 2)}{x^{2}} = 0$

Factor the polynomial by factoring out the greatest common factor, $- 17 x + 2$ .

$\frac{(- 17 x + 2) (x + 3)}{x^{2}} = 0$

Step 10

Set the numerator equal to zero.

$(- 17 x + 2) (x + 3) = 0$

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 $0$ .

$- 17 x + 2 = 0$

$x + 3 = 0$

Set $- 17 x + 2$ equal to $0$ and solve for $x$ .

Set $- 17 x + 2$ equal to $0$ .

$- 17 x + 2 = 0$

Solve $- 17 x + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

$- 17 x = - 2$

Divide each term in $- 17 x = - 2$ by $- 17$ and simplify.

Divide each term in $- 17 x = - 2$ by $- 17$ .

$\frac{- 17 x}{- 17} = \frac{- 2}{- 17}$

Simplify the left side.

Cancel the common factor of $- 17$ .

Cancel the common factor.

$\frac{- 17 x}{- 17} = \frac{- 2}{- 17}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 17}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{2}{17}$

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

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

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $(- 17 x + 2) (x + 3) = 0$ true.

$x = \frac{2}{17}, - 3$

Solve by Factoring 6/(x^2)-49/x=17

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