Linear Algebra Examples

$\frac{\frac{x^{2} - 81}{3 x - 18}}{\frac{x^{2} + 18 x + 81}{x^{2} + 3 x - 54}}$

Step 1

Set the denominator in $\frac{x^{2} - 81}{3 x - 18}$ equal to $0$ to find where the expression is undefined.

$3 x - 18 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $18$ to both sides of the equation.

$3 x = 18$

Step 2.2

Divide each term in $3 x = 18$ by $3$ and simplify.

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Step 2.2.1

Divide each term in $3 x = 18$ by $3$ .

$\frac{3 x}{3} = \frac{18}{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $3$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{18}{3}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $18$ by $3$ .

$x = 6$

Step 3

Set the denominator in $\frac{x^{2} + 18 x + 81}{x^{2} + 3 x - 54}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 3 x - 54 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Factor $x^{2} + 3 x - 54$ using the AC method.

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Step 4.1.1

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 $- 54$ and whose sum is $3$ .

$- 6, 9$

Step 4.1.2

Write the factored form using these integers.

$(x - 6) (x + 9) = 0$

Step 4.2

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

$x - 6 = 0$

$x + 9 = 0$

Step 4.3

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 4.3.2

Add $6$ to both sides of the equation.

$x = 6$

Step 4.4

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 4.4.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 4.5

The final solution is all the values that make $(x - 6) (x + 9) = 0$ true.

$x = 6, - 9$

Step 5

Set the denominator in $\frac{\frac{x^{2} - 81}{3 x - 18}}{\frac{x^{2} + 18 x + 81}{x^{2} + 3 x - 54}}$ equal to $0$ to find where the expression is undefined.

$\frac{x^{2} + 18 x + 81}{x^{2} + 3 x - 54} = 0$

Step 6

Solve for $x$ .

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Step 6.1

Set the numerator equal to zero.

$x^{2} + 18 x + 81 = 0$

Step 6.2

Solve the equation for $x$ .

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Step 6.2.1

Factor using the perfect square rule.

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Step 6.2.1.1

Rewrite $81$ as $9^{2}$ .

$x^{2} + 18 x + 9^{2} = 0$

Step 6.2.1.2

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

$18 x = 2 \cdot x \cdot 9$

Step 6.2.1.3

Rewrite the polynomial.

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

Step 6.2.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 9$ .

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

Step 6.2.2

Set the $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 6.2.3

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 6.3

Exclude the solutions that do not make $\frac{x^{2} + 18 x + 81}{x^{2} + 3 x - 54} = 0$ true.

$No solution$

Step 7

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 9) \cup (- 9, 6) \cup (6, \infty)$

Set-Builder Notation:

${x | x \neq 6, - 9}$

Step 8