Precalculus Examples

$2 - \frac{1}{x} - \frac{9}{x^{2}} = 0$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, x^{2}, 1$

Step 1.2

Since $1, x, x^{2}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1, 1, 1$ then find LCM for the variable part $x^{1}, x^{2}$ .

Step 1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.5

The LCM of $1, 1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 1.6

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

Step 1.7

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 1.8

The LCM of $x^{1}, x^{2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x$

Step 1.9

Multiply $x$ by $x$ .

$x^{2}$

Step 2

Multiply each term in $2 - \frac{1}{x} - \frac{9}{x^{2}} = 0$ by $x^{2}$ to eliminate the fractions.

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Step 2.1

Multiply each term in $2 - \frac{1}{x} - \frac{9}{x^{2}} = 0$ by $x^{2}$ .

$2 x^{2} - \frac{1}{x} x^{2} - \frac{9}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $x$ .

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Step 2.2.1.1.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 2.2.1.1.2

Factor $x$ out of $x^{2}$ .

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

Step 2.2.1.1.3

Cancel the common factor.

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

Step 2.2.1.1.4

Rewrite the expression.

$2 x^{2} - x - \frac{9}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.1.2

Cancel the common factor of $x^{2}$ .

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Step 2.2.1.2.1

Move the leading negative in $- \frac{9}{x^{2}}$ into the numerator.

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

Step 2.2.1.2.2

Cancel the common factor.

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

Step 2.2.1.2.3

Rewrite the expression.

$2 x^{2} - x - 9 = 0 x^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $0$ by $x^{2}$ .

$2 x^{2} - x - 9 = 0$

Step 3

Solve the equation.

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Step 3.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.2

Substitute the values $a = 2$ , $b = - 1$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (2 \cdot - 9)}}{2 \cdot 2}$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot - 9}}{2 \cdot 2}$

Step 3.3.1.2

Multiply $- 4 \cdot 2 \cdot - 9$ .

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Step 3.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{1 \pm \sqrt{1 - 8 \cdot - 9}}{2 \cdot 2}$

Step 3.3.1.2.2

Multiply $- 8$ by $- 9$ .

$x = \frac{1 \pm \sqrt{1 + 72}}{2 \cdot 2}$

Step 3.3.1.3

Add $1$ and $72$ .

$x = \frac{1 \pm \sqrt{73}}{2 \cdot 2}$

Step 3.3.2

Multiply $2$ by $2$ .

$x = \frac{1 \pm \sqrt{73}}{4}$

Step 3.4

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{73}}{4}, \frac{1 - \sqrt{73}}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{73}}{4}, \frac{1 - \sqrt{73}}{4}$

Decimal Form:

$x = 2.38600093 \dots, - 1.88600093 \dots$