Precalculus Examples

$1 = \frac{6 x^{2}}{x^{4} + 9}$

Step 1

Rewrite the equation as $\frac{6 x^{2}}{x^{4} + 9} = 1$ .

$\frac{6 x^{2}}{x^{4} + 9} = 1$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$x^{4} + 9, 1$

Step 2.2

Remove parentheses.

$x^{4} + 9, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x^{4} + 9$

Step 3

Multiply each term in $\frac{6 x^{2}}{x^{4} + 9} = 1$ by $x^{4} + 9$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{6 x^{2}}{x^{4} + 9} = 1$ by $x^{4} + 9$ .

$\frac{6 x^{2}}{x^{4} + 9} (x^{4} + 9) = 1 (x^{4} + 9)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x^{4} + 9$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{6 x^{2}}{x^{4} + 9} (x^{4} + 9) = 1 (x^{4} + 9)$

Step 3.2.1.2

Rewrite the expression.

$6 x^{2} = 1 (x^{4} + 9)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $x^{4} + 9$ by $1$ .

$6 x^{2} = x^{4} + 9$

Step 4

Solve the equation.

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Step 4.1

Subtract $x^{4}$ from both sides of the equation.

$6 x^{2} - x^{4} = 9$

Step 4.2

Subtract $9$ from both sides of the equation.

$6 x^{2} - x^{4} - 9 = 0$

Step 4.3

Factor the left side of the equation.

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Step 4.3.1

Factor $- 1$ out of $6 x^{2} - x^{4} - 9$ .

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Step 4.3.1.1

Reorder $6 x^{2}$ and $- x^{4}$ .

$- x^{4} + 6 x^{2} - 9 = 0$

Step 4.3.1.2

Factor $- 1$ out of $- x^{4}$ .

$- (x^{4}) + 6 x^{2} - 9 = 0$

Step 4.3.1.3

Factor $- 1$ out of $6 x^{2}$ .

$- (x^{4}) - (- 6 x^{2}) - 9 = 0$

Step 4.3.1.4

Rewrite $- 9$ as $- 1 (9)$ .

$- (x^{4}) - (- 6 x^{2}) - 1 \cdot 9 = 0$

Step 4.3.1.5

Factor $- 1$ out of $- (x^{4}) - (- 6 x^{2})$ .

$- (x^{4} - 6 x^{2}) - 1 \cdot 9 = 0$

Step 4.3.1.6

Factor $- 1$ out of $- (x^{4} - 6 x^{2}) - 1 (9)$ .

$- (x^{4} - 6 x^{2} + 9) = 0$

Step 4.3.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- ({(x^{2})}^{2} - 6 x^{2} + 9) = 0$

Step 4.3.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- (u^{2} - 6 u + 9) = 0$

Step 4.3.4

Factor using the perfect square rule.

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Step 4.3.4.1

Rewrite $9$ as $3^{2}$ .

$- (u^{2} - 6 u + 3^{2}) = 0$

Step 4.3.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 u = 2 \cdot u \cdot 3$

Step 4.3.4.3

Rewrite the polynomial.

$- (u^{2} - 2 \cdot u \cdot 3 + 3^{2}) = 0$

Step 4.3.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 3$ .

$- {(u - 3)}^{2} = 0$

Step 4.3.5

Replace all occurrences of $u$ with $x^{2}$ .

$- {(x^{2} - 3)}^{2} = 0$

Step 4.4

Divide each term in $- {(x^{2} - 3)}^{2} = 0$ by $- 1$ and simplify.

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Step 4.4.1

Divide each term in $- {(x^{2} - 3)}^{2} = 0$ by $- 1$ .

$\frac{- {(x^{2} - 3)}^{2}}{- 1} = \frac{0}{- 1}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Dividing two negative values results in a positive value.

$\frac{{(x^{2} - 3)}^{2}}{1} = \frac{0}{- 1}$

Step 4.4.2.2

Divide ${(x^{2} - 3)}^{2}$ by $1$ .

${(x^{2} - 3)}^{2} = \frac{0}{- 1}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Divide $0$ by $- 1$ .

${(x^{2} - 3)}^{2} = 0$

Step 4.5

Set the $x^{2} - 3$ equal to $0$ .

$x^{2} - 3 = 0$

Step 4.6

Solve for $x$ .

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Step 4.6.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 4.6.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{3}$

Step 4.6.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.6.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{3}$

Step 4.6.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{3}$

Step 4.6.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{3}, - \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{3}, - \sqrt{3}$

Decimal Form:

$x = 1.73205080 \dots, - 1.73205080 \dots$