Algebra Examples

$1 = \frac{8}{a - 3} - \frac{48}{a^{2} - 9}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{8}{a - 3}$ from both sides of the equation.

$1 - \frac{8}{a - 3} = - \frac{48}{a^{2} - 9}$

Step 1.2

Add $\frac{48}{a^{2} - 9}$ to both sides of the equation.

$1 - \frac{8}{a - 3} + \frac{48}{a^{2} - 9} = 0$

Step 2

Simplify $1 - \frac{8}{a - 3} + \frac{48}{a^{2} - 9}$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

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

$1 - \frac{8}{a - 3} + \frac{48}{a^{2} - 3^{2}} = 0$

Step 2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 3$ .

$1 - \frac{8}{a - 3} + \frac{48}{(a + 3) (a - 3)} = 0$

Step 2.2

Write $1$ as a fraction with a common denominator.

$\frac{a - 3}{a - 3} - \frac{8}{a - 3} + \frac{48}{(a + 3) (a - 3)} = 0$

Step 2.3

Combine the numerators over the common denominator.

$\frac{a - 3 - 8}{a - 3} + \frac{48}{(a + 3) (a - 3)} = 0$

Step 2.4

Subtract $8$ from $- 3$ .

$\frac{a - 11}{a - 3} + \frac{48}{(a + 3) (a - 3)} = 0$

Step 2.5

To write $\frac{a - 11}{a - 3}$ as a fraction with a common denominator, multiply by $\frac{a + 3}{a + 3}$ .

$\frac{a - 11}{a - 3} \cdot \frac{a + 3}{a + 3} + \frac{48}{(a + 3) (a - 3)} = 0$

Step 2.6

Write each expression with a common denominator of $(a - 3) (a + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.6.1

Multiply $\frac{a - 11}{a - 3}$ by $\frac{a + 3}{a + 3}$ .

$\frac{(a - 11) (a + 3)}{(a - 3) (a + 3)} + \frac{48}{(a + 3) (a - 3)} = 0$

Step 2.6.2

Reorder the factors of $(a - 3) (a + 3)$ .

$\frac{(a - 11) (a + 3)}{(a + 3) (a - 3)} + \frac{48}{(a + 3) (a - 3)} = 0$

Step 2.7

Combine the numerators over the common denominator.

$\frac{(a - 11) (a + 3) + 48}{(a + 3) (a - 3)} = 0$

Step 2.8

Simplify the numerator.

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Step 2.8.1

Expand $(a - 11) (a + 3)$ using the FOIL Method.

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Step 2.8.1.1

Apply the distributive property.

$\frac{a (a + 3) - 11 (a + 3) + 48}{(a + 3) (a - 3)} = 0$

Step 2.8.1.2

Apply the distributive property.

$\frac{a \cdot a + a \cdot 3 - 11 (a + 3) + 48}{(a + 3) (a - 3)} = 0$

Step 2.8.1.3

Apply the distributive property.

$\frac{a \cdot a + a \cdot 3 - 11 a - 11 \cdot 3 + 48}{(a + 3) (a - 3)} = 0$

Step 2.8.2

Simplify and combine like terms.

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Step 2.8.2.1

Simplify each term.

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Step 2.8.2.1.1

Multiply $a$ by $a$ .

$\frac{a^{2} + a \cdot 3 - 11 a - 11 \cdot 3 + 48}{(a + 3) (a - 3)} = 0$

Step 2.8.2.1.2

Move $3$ to the left of $a$ .

$\frac{a^{2} + 3 \cdot a - 11 a - 11 \cdot 3 + 48}{(a + 3) (a - 3)} = 0$

Step 2.8.2.1.3

Multiply $- 11$ by $3$ .

$\frac{a^{2} + 3 a - 11 a - 33 + 48}{(a + 3) (a - 3)} = 0$

Step 2.8.2.2

Subtract $11 a$ from $3 a$ .

$\frac{a^{2} - 8 a - 33 + 48}{(a + 3) (a - 3)} = 0$

Step 2.8.3

Add $- 33$ and $48$ .

$\frac{a^{2} - 8 a + 15}{(a + 3) (a - 3)} = 0$

Step 2.8.4

Factor $a^{2} - 8 a + 15$ using the AC method.

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Step 2.8.4.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 2.8.4.2

Write the factored form using these integers.

$\frac{(a - 5) (a - 3)}{(a + 3) (a - 3)} = 0$

Step 2.9

Cancel the common factor of $a - 3$ .

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Step 2.9.1

Cancel the common factor.

$\frac{(a - 5) (a - 3)}{(a + 3) (a - 3)} = 0$

Step 2.9.2

Rewrite the expression.

$\frac{a - 5}{a + 3} = 0$

Step 3

Set the numerator equal to zero.

$a - 5 = 0$

Step 4

Add $5$ to both sides of the equation.

$a = 5$