Solve by Factoring 5/(x^2)-43/x=18

$\frac{5}{x^{2}} - \frac{43}{x} = 18$

Step 1

Subtract $18$ from both sides of the equation.

$\frac{5}{x^{2}} - \frac{43}{x} - 18 = 0$

Step 2

Reorder terms.

$- \frac{43}{x} + \frac{5}{x^{2}} - 18 = 0$

Step 3

To write $- \frac{43}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{43}{x} \cdot \frac{x}{x} + \frac{5}{x^{2}} - 18 = 0$

Step 4

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

Multiply $\frac{43}{x}$ by $\frac{x}{x}$ .

$- \frac{43 x}{x \cdot x} + \frac{5}{x^{2}} - 18 = 0$

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

$- \frac{43 x}{x \cdot x} + \frac{5}{x^{2}} - 18 = 0$

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

$- \frac{43 x}{x \cdot x} + \frac{5}{x^{2}} - 18 = 0$

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

$- \frac{43 x}{x^{1 + 1}} + \frac{5}{x^{2}} - 18 = 0$

Add $1$ and $1$ .

$- \frac{43 x}{x^{2}} + \frac{5}{x^{2}} - 18 = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{- 43 x + 5}{x^{2}} - 18 = 0$

Step 6

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{- 43 x + 5}{x^{2}} - 18 \cdot \frac{x^{2}}{x^{2}} = 0$

Step 7

Combine $- 18$ and $\frac{x^{2}}{x^{2}}$ .

$\frac{- 43 x + 5}{x^{2}} + \frac{- 18 x^{2}}{x^{2}} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{- 43 x + 5 - 18 x^{2}}{x^{2}} = 0$

Step 9

Factor by grouping.

Reorder terms.

$\frac{- 18 x^{2} - 43 x + 5}{x^{2}} = 0$

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 $a \cdot c = - 18 \cdot 5 = - 90$ and whose sum is $b = - 43$ .

Factor $- 43$ out of $- 43 x$ .

$\frac{- 18 x^{2} - 43 x + 5}{x^{2}} = 0$

Rewrite $- 43$ as $2$ plus $- 45$

$\frac{- 18 x^{2} + (2 - 45) x + 5}{x^{2}} = 0$

Apply the distributive property.

$\frac{- 18 x^{2} + 2 x - 45 x + 5}{x^{2}} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- 18 x^{2} + 2 x) - 45 x + 5}{x^{2}} = 0$

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

$\frac{2 x (- 9 x + 1) + 5 (- 9 x + 1)}{x^{2}} = 0$

Factor the polynomial by factoring out the greatest common factor, $- 9 x + 1$ .

$\frac{(- 9 x + 1) (2 x + 5)}{x^{2}} = 0$

Step 10

Set the numerator equal to zero.

$(- 9 x + 1) (2 x + 5) = 0$

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 $0$ .

$- 9 x + 1 = 0$

$2 x + 5 = 0$

Set $- 9 x + 1$ equal to $0$ and solve for $x$ .

Set $- 9 x + 1$ equal to $0$ .

$- 9 x + 1 = 0$

Solve $- 9 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$- 9 x = - 1$

Divide each term in $- 9 x = - 1$ by $- 9$ and simplify.

Divide each term in $- 9 x = - 1$ by $- 9$ .

$\frac{- 9 x}{- 9} = \frac{- 1}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 x}{- 9} = \frac{- 1}{- 9}$

Divide $x$ by $1$ .

$x = \frac{- 1}{- 9}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{1}{9}$

Set $2 x + 5$ equal to $0$ and solve for $x$ .

Set $2 x + 5$ equal to $0$ .

$2 x + 5 = 0$

Solve $2 x + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Divide each term in $2 x = - 5$ by $2$ and simplify.

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

The final solution is all the values that make $(- 9 x + 1) (2 x + 5) = 0$ true.

$x = \frac{1}{9}, - \frac{5}{2}$

Solve by Factoring 5/(x^2)-43/x=18

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