Algebra Examples

$\frac{5}{x^{2}} - \frac{54}{x} = 11$

Step 1

Subtract $11$ from both sides of the equation.

$\frac{5}{x^{2}} - \frac{54}{x} - 11 = 0$

Step 2

Reorder terms.

$- \frac{54}{x} + \frac{5}{x^{2}} - 11 = 0$

Step 3

To write $- \frac{54}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{54}{x} \cdot \frac{x}{x} + \frac{5}{x^{2}} - 11 = 0$

Step 4

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

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Step 4.1

Multiply $\frac{54}{x}$ by $\frac{x}{x}$ .

$- \frac{54 x}{x \cdot x} + \frac{5}{x^{2}} - 11 = 0$

Step 4.2

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

$- \frac{54 x}{x \cdot x} + \frac{5}{x^{2}} - 11 = 0$

Step 4.3

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

$- \frac{54 x}{x \cdot x} + \frac{5}{x^{2}} - 11 = 0$

Step 4.4

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

$- \frac{54 x}{x^{1 + 1}} + \frac{5}{x^{2}} - 11 = 0$

Step 4.5

Add $1$ and $1$ .

$- \frac{54 x}{x^{2}} + \frac{5}{x^{2}} - 11 = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{- 54 x + 5}{x^{2}} - 11 = 0$

Step 6

To write $- 11$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$\frac{- 54 x + 5}{x^{2}} - 11 \cdot \frac{x^{2}}{x^{2}} = 0$

Step 7

Combine $- 11$ and $\frac{x^{2}}{x^{2}}$ .

$\frac{- 54 x + 5}{x^{2}} + \frac{- 11 x^{2}}{x^{2}} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{- 54 x + 5 - 11 x^{2}}{x^{2}} = 0$

Step 9

Factor by grouping.

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Step 9.1

Reorder terms.

$\frac{- 11 x^{2} - 54 x + 5}{x^{2}} = 0$

Step 9.2

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 $a \cdot c = - 11 \cdot 5 = - 55$ and whose sum is $b = - 54$ .

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Step 9.2.1

Factor $- 54$ out of $- 54 x$ .

$\frac{- 11 x^{2} - 54 x + 5}{x^{2}} = 0$

Step 9.2.2

Rewrite $- 54$ as $1$ plus $- 55$

$\frac{- 11 x^{2} + (1 - 55) x + 5}{x^{2}} = 0$

Step 9.2.3

Apply the distributive property.

$\frac{- 11 x^{2} + 1 x - 55 x + 5}{x^{2}} = 0$

Step 9.2.4

Multiply $x$ by $1$ .

$\frac{- 11 x^{2} + x - 55 x + 5}{x^{2}} = 0$

Step 9.3

Factor out the greatest common factor from each group.

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Step 9.3.1

Group the first two terms and the last two terms.

$\frac{(- 11 x^{2} + x) - 55 x + 5}{x^{2}} = 0$

Step 9.3.2

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

$\frac{x (- 11 x + 1) + 5 (- 11 x + 1)}{x^{2}} = 0$

Step 9.4

Factor the polynomial by factoring out the greatest common factor, $- 11 x + 1$ .

$\frac{(- 11 x + 1) (x + 5)}{x^{2}} = 0$

Step 10

Set the numerator equal to zero.

$(- 11 x + 1) (x + 5) = 0$

Step 11

Solve the equation for $x$ .

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Step 11.1

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

$- 11 x + 1 = 0$

$x + 5 = 0$

Step 11.2

Set $- 11 x + 1$ equal to $0$ and solve for $x$ .

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Step 11.2.1

Set $- 11 x + 1$ equal to $0$ .

$- 11 x + 1 = 0$

Step 11.2.2

Solve $- 11 x + 1 = 0$ for $x$ .

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Step 11.2.2.1

Subtract $1$ from both sides of the equation.

$- 11 x = - 1$

Step 11.2.2.2

Divide each term in $- 11 x = - 1$ by $- 11$ and simplify.

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Step 11.2.2.2.1

Divide each term in $- 11 x = - 1$ by $- 11$ .

$\frac{- 11 x}{- 11} = \frac{- 1}{- 11}$

Step 11.2.2.2.2

Simplify the left side.

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Step 11.2.2.2.2.1

Cancel the common factor of $- 11$ .

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Step 11.2.2.2.2.1.1

Cancel the common factor.

$\frac{- 11 x}{- 11} = \frac{- 1}{- 11}$

Step 11.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 11}$

Step 11.2.2.2.3

Simplify the right side.

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Step 11.2.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{11}$

Step 11.3

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

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Step 11.3.1

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

$x + 5 = 0$

Step 11.3.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 11.4

The final solution is all the values that make $(- 11 x + 1) (x + 5) = 0$ true.

$x = \frac{1}{11}, - 5$