Finite Math Examples

$x = \frac{x^{2} - 36}{x^{2} - 9 x + 18}$

Step 1

Factor each term.

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Step 1.1

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

$x = \frac{x^{2} - 6^{2}}{x^{2} - 9 x + 18}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$x = \frac{(x + 6) (x - 6)}{x^{2} - 9 x + 18}$

Step 1.3

Factor $x^{2} - 9 x + 18$ using the AC method.

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Step 1.3.1

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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Step 1.3.2

Write the factored form using these integers.

$x = \frac{(x + 6) (x - 6)}{(x - 6) (x - 3)}$

Step 1.4

Reduce the expression $\frac{(x + 6) (x - 6)}{(x - 6) (x - 3)}$ by cancelling the common factors.

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Step 1.4.1

Cancel the common factor.

$x = \frac{(x + 6) (x - 6)}{(x - 6) (x - 3)}$

Step 1.4.2

Rewrite the expression.

$x = \frac{x + 6}{x - 3}$

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.

$1, x - 3$

Step 2.2

Remove parentheses.

$1, x - 3$

Step 2.3

The LCM of one and any expression is the expression.

$x - 3$

Step 3

Multiply each term in $x = \frac{x + 6}{x - 3}$ by $x - 3$ to eliminate the fractions.

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Step 3.1

Multiply each term in $x = \frac{x + 6}{x - 3}$ by $x - 3$ .

$x (x - 3) = \frac{x + 6}{x - 3} (x - 3)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$x \cdot x + x \cdot - 3 = \frac{x + 6}{x - 3} (x - 3)$

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 3 = \frac{x + 6}{x - 3} (x - 3)$

Step 3.2.2.2

Move $- 3$ to the left of $x$ .

$x^{2} - 3 x = \frac{x + 6}{x - 3} (x - 3)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $x - 3$ .

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Step 3.3.1.1

Cancel the common factor.

$x^{2} - 3 x = \frac{x + 6}{x - 3} (x - 3)$

Step 3.3.1.2

Rewrite the expression.

$x^{2} - 3 x = x + 6$

Step 4

Solve the equation.

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Step 4.1

Move all terms containing $x$ to the left side of the equation.

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Step 4.1.1

Subtract $x$ from both sides of the equation.

$x^{2} - 3 x - x = 6$

Step 4.1.2

Subtract $x$ from $- 3 x$ .

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

Step 4.2

Subtract $6$ from both sides of the equation.

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

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot - 6)}}{2 \cdot 1}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot - 6$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 6}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $- 6$ .

$x = \frac{4 \pm \sqrt{16 + 24}}{2 \cdot 1}$

Step 4.5.1.3

Add $16$ and $24$ .

$x = \frac{4 \pm \sqrt{40}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 4.5.1.4.1

Factor $4$ out of $40$ .

$x = \frac{4 \pm \sqrt{4 (10)}}{2 \cdot 1}$

Step 4.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Step 4.5.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{10}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{10}}{2}$

Step 4.5.3

Simplify $\frac{4 \pm 2 \sqrt{10}}{2}$ .

$x = 2 \pm \sqrt{10}$

Step 4.6

The final answer is the combination of both solutions.

$x = 2 + \sqrt{10}, 2 - \sqrt{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 2 + \sqrt{10}, 2 - \sqrt{10}$

Decimal Form:

$x = 5.16227766 \dots, - 1.16227766 \dots$