Precalculus Examples

$x^{2} y = 9$ , $x^{2} + 4 y + 12 = 0$

Step 1

Divide each term in $x^{2} y = 9$ by $x^{2}$ and simplify.

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Step 1.1

Divide each term in $x^{2} y = 9$ by $x^{2}$ .

$\frac{x^{2} y}{x^{2}} = \frac{9}{x^{2}}$

$x^{2} + 4 y + 12 = 0$

Step 1.2

Simplify the left side.

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Step 1.2.1

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

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Step 1.2.1.1

Cancel the common factor.

$\frac{x^{2} y}{x^{2}} = \frac{9}{x^{2}}$

$x^{2} + 4 y + 12 = 0$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{9}{x^{2}}$

$x^{2} + 4 y + 12 = 0$

$y = \frac{9}{x^{2}}$

$x^{2} + 4 y + 12 = 0$

$y = \frac{9}{x^{2}}$

$x^{2} + 4 y + 12 = 0$

$y = \frac{9}{x^{2}}$

$x^{2} + 4 y + 12 = 0$

Step 2

Replace all occurrences of $y$ with $\frac{9}{x^{2}}$ in each equation.

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Step 2.1

Replace all occurrences of $y$ in $x^{2} + 4 y + 12 = 0$ with $\frac{9}{x^{2}}$ .

$x^{2} + 4 (\frac{9}{x^{2}}) + 12 = 0$

$y = \frac{9}{x^{2}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Multiply $4 (\frac{9}{x^{2}})$ .

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Step 2.2.1.1

Combine $4$ and $\frac{9}{x^{2}}$ .

$x^{2} + \frac{4 \cdot 9}{x^{2}} + 12 = 0$

$y = \frac{9}{x^{2}}$

Step 2.2.1.2

Multiply $4$ by $9$ .

$x^{2} + \frac{36}{x^{2}} + 12 = 0$

$y = \frac{9}{x^{2}}$

$x^{2} + \frac{36}{x^{2}} + 12 = 0$

$y = \frac{9}{x^{2}}$

$x^{2} + \frac{36}{x^{2}} + 12 = 0$

$y = \frac{9}{x^{2}}$

$x^{2} + \frac{36}{x^{2}} + 12 = 0$

$y = \frac{9}{x^{2}}$

Step 3

Solve for $x$ in $x^{2} + \frac{36}{x^{2}} + 12 = 0$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

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

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

$y = \frac{9}{x^{2}}$

Step 3.1.2

The LCM of one and any expression is the expression.

$x^{2}$

$y = \frac{9}{x^{2}}$

$x^{2}$

$y = \frac{9}{x^{2}}$

Step 3.2

Multiply each term in $x^{2} + \frac{36}{x^{2}} + 12 = 0$ by $x^{2}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $x^{2} + \frac{36}{x^{2}} + 12 = 0$ by $x^{2}$ .

$x^{2} x^{2} + \frac{36}{x^{2}} \cdot x^{2} + 12 x^{2} = 0 x^{2}$

$y = \frac{9}{x^{2}}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.2.2.1.1.1

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

$x^{2 + 2} + \frac{36}{x^{2}} \cdot x^{2} + 12 x^{2} = 0 x^{2}$

$y = \frac{9}{x^{2}}$

Step 3.2.2.1.1.2

Add $2$ and $2$ .

$x^{4} + \frac{36}{x^{2}} \cdot x^{2} + 12 x^{2} = 0 x^{2}$

$y = \frac{9}{x^{2}}$

$x^{4} + \frac{36}{x^{2}} \cdot x^{2} + 12 x^{2} = 0 x^{2}$

$y = \frac{9}{x^{2}}$

Step 3.2.2.1.2

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

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Step 3.2.2.1.2.1

Cancel the common factor.

$x^{4} + \frac{36}{x^{2}} \cdot x^{2} + 12 x^{2} = 0 x^{2}$

$y = \frac{9}{x^{2}}$

Step 3.2.2.1.2.2

Rewrite the expression.

$x^{4} + 36 + 12 x^{2} = 0 x^{2}$

$y = \frac{9}{x^{2}}$

$x^{4} + 36 + 12 x^{2} = 0 x^{2}$

$y = \frac{9}{x^{2}}$

$x^{4} + 36 + 12 x^{2} = 0 x^{2}$

$y = \frac{9}{x^{2}}$

$x^{4} + 36 + 12 x^{2} = 0 x^{2}$

$y = \frac{9}{x^{2}}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

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

$x^{4} + 36 + 12 x^{2} = 0$

$y = \frac{9}{x^{2}}$

$x^{4} + 36 + 12 x^{2} = 0$

$y = \frac{9}{x^{2}}$

$x^{4} + 36 + 12 x^{2} = 0$

$y = \frac{9}{x^{2}}$

Step 3.3

Solve the equation.

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Step 3.3.1

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} + 12 u + 36 = 0$

$u = x^{2}$

$y = \frac{9}{x^{2}}$

Step 3.3.2

Factor using the perfect square rule.

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Step 3.3.2.1

Rewrite $36$ as $6^{2}$ .

$u^{2} + 12 u + 6^{2} = 0$

$y = \frac{9}{x^{2}}$

Step 3.3.2.2

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

$12 u = 2 \cdot u \cdot 6$

$y = \frac{9}{x^{2}}$

Step 3.3.2.3

Rewrite the polynomial.

$u^{2} + 2 \cdot u \cdot 6 + 6^{2} = 0$

$y = \frac{9}{x^{2}}$

Step 3.3.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 6$ .

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

$y = \frac{9}{x^{2}}$

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

$y = \frac{9}{x^{2}}$

Step 3.3.3

Set the $u + 6$ equal to $0$ .

$u + 6 = 0$

$y = \frac{9}{x^{2}}$

Step 3.3.4

Subtract $6$ from both sides of the equation.

$u = - 6$

$y = \frac{9}{x^{2}}$

Step 3.3.5

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = - 6$

$y = \frac{9}{x^{2}}$

Step 3.3.6

Solve the equation for $x$ .

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Step 3.3.6.1

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

$x = \pm \sqrt{- 6}$

$y = \frac{9}{x^{2}}$

Step 3.3.6.2

Simplify $\pm \sqrt{- 6}$ .

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Step 3.3.6.2.1

Rewrite $- 6$ as $- 1 (6)$ .

$x = \pm \sqrt{- 1 \cdot 6}$

$y = \frac{9}{x^{2}}$

Step 3.3.6.2.2

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{6}$

$y = \frac{9}{x^{2}}$

Step 3.3.6.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \sqrt{6}$

$y = \frac{9}{x^{2}}$

$x = \pm i \sqrt{6}$

$y = \frac{9}{x^{2}}$

Step 3.3.6.3

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

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Step 3.3.6.3.1

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

$x = i \sqrt{6}$

$y = \frac{9}{x^{2}}$

Step 3.3.6.3.2

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

$x = - i \sqrt{6}$

$y = \frac{9}{x^{2}}$

Step 3.3.6.3.3

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