Algebra Examples

$x^{- 2} + 10 x^{- 1} + 16 = 0$

Step 1

Factor $x^{- 2} + 10 x^{- 1} + 16$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $16$ and whose sum is $10$ .

$2, 8$

Step 1.2

Write the factored form using these integers.

$(x^{- 1} + 2) (x^{- 1} + 8) = 0$

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(\frac{1}{x} + 2) (x^{- 1} + 8) = 0$

Step 3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(\frac{1}{x} + 2) (\frac{1}{x} + 8) = 0$

Step 4

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

$(\frac{1}{x} + \frac{2 x}{x}) (\frac{1}{x} + 8) = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{1 + 2 x}{x} \cdot (\frac{1}{x} + 8) = 0$

Step 6

To write $8$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1 + 2 x}{x} \cdot (\frac{1}{x} + \frac{8 x}{x}) = 0$

Step 7

Combine the numerators over the common denominator.

$\frac{1 + 2 x}{x} \cdot \frac{1 + 8 x}{x} = 0$

Step 8

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

$\frac{1 + 2 x}{x} = 0$

$\frac{1 + 8 x}{x} = 0$

Step 9

Set $\frac{1 + 2 x}{x}$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $\frac{1 + 2 x}{x}$ equal to $0$ .

$\frac{1 + 2 x}{x} = 0$

Step 9.2

Solve $\frac{1 + 2 x}{x} = 0$ for $x$ .

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Step 9.2.1

Set the numerator equal to zero.

$1 + 2 x = 0$

Step 9.2.2

Solve the equation for $x$ .

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Step 9.2.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 9.2.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 9.2.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 9.2.2.2.2

Simplify the left side.

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Step 9.2.2.2.2.1

Cancel the common factor of $2$ .

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Step 9.2.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 9.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 9.2.2.2.3

Simplify the right side.

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Step 9.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 10

Set $\frac{1 + 8 x}{x}$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $\frac{1 + 8 x}{x}$ equal to $0$ .

$\frac{1 + 8 x}{x} = 0$

Step 10.2

Solve $\frac{1 + 8 x}{x} = 0$ for $x$ .

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Step 10.2.1

Set the numerator equal to zero.

$1 + 8 x = 0$

Step 10.2.2

Solve the equation for $x$ .

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Step 10.2.2.1

Subtract $1$ from both sides of the equation.

$8 x = - 1$

Step 10.2.2.2

Divide each term in $8 x = - 1$ by $8$ and simplify.

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Step 10.2.2.2.1

Divide each term in $8 x = - 1$ by $8$ .

$\frac{8 x}{8} = \frac{- 1}{8}$

Step 10.2.2.2.2

Simplify the left side.

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Step 10.2.2.2.2.1

Cancel the common factor of $8$ .

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Step 10.2.2.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 1}{8}$

Step 10.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{8}$

Step 10.2.2.2.3

Simplify the right side.

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Step 10.2.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{8}$

Step 11

The final solution is all the values that make $\frac{1 + 2 x}{x} \cdot \frac{1 + 8 x}{x} = 0$ true.

$x = - \frac{1}{2}, - \frac{1}{8}$