Algebra Examples

$\frac{15}{a^{2} - 1} = \frac{5}{2 a - 2}$

Step 1

Subtract $\frac{5}{2 a - 2}$ from both sides of the equation.

$\frac{15}{a^{2} - 1} - \frac{5}{2 a - 2} = 0$

Step 2

Simplify each term.

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Rewrite $1$ as $1^{2}$ .

$\frac{15}{a^{2} - 1^{2}} - \frac{5}{2 a - 2} = 0$

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 $b = 1$ .

$\frac{15}{(a + 1) (a - 1)} - \frac{5}{2 a - 2} = 0$

Step 2.2

Factor $2$ out of $2 a - 2$ .

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Step 2.2.1

Factor $2$ out of $2 a$ .

$\frac{15}{(a + 1) (a - 1)} - \frac{5}{2 (a) - 2} = 0$

Step 2.2.2

Factor $2$ out of $- 2$ .

$\frac{15}{(a + 1) (a - 1)} - \frac{5}{2 (a) + 2 (- 1)} = 0$

Step 2.2.3

Factor $2$ out of $2 (a) + 2 (- 1)$ .

$\frac{15}{(a + 1) (a - 1)} - \frac{5}{2 (a - 1)} = 0$

Step 3

Set the denominator in $\frac{15}{(a + 1) (a - 1)}$ equal to $0$ to find where the expression is undefined.

$(a + 1) (a - 1) = 0$

Step 4

Solve for $a$ .

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Step 4.1

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

$a + 1 = 0$

$a - 1 = 0$

Step 4.2

Set $a + 1$ equal to $0$ and solve for $a$ .

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Step 4.2.1

Set $a + 1$ equal to $0$ .

$a + 1 = 0$

Step 4.2.2

Subtract $1$ from both sides of the equation.

$a = - 1$

Step 4.3

Set $a - 1$ equal to $0$ and solve for $a$ .

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Step 4.3.1

Set $a - 1$ equal to $0$ .

$a - 1 = 0$

Step 4.3.2

Add $1$ to both sides of the equation.

$a = 1$

Step 4.4

The final solution is all the values that make $(a + 1) (a - 1) = 0$ true.

$a = - 1, 1$

Step 5

Set the denominator in $\frac{5}{2 (a - 1)}$ equal to $0$ to find where the expression is undefined.

$2 (a - 1) = 0$

Step 6

Solve for $a$ .

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Step 6.1

Divide each term in $2 (a - 1) = 0$ by $2$ and simplify.

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Step 6.1.1

Divide each term in $2 (a - 1) = 0$ by $2$ .

$\frac{2 (a - 1)}{2} = \frac{0}{2}$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $2$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{2 (a - 1)}{2} = \frac{0}{2}$

Step 6.1.2.1.2

Divide $a - 1$ by $1$ .

$a - 1 = \frac{0}{2}$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Divide $0$ by $2$ .

$a - 1 = 0$

Step 6.2

Add $1$ to both sides of the equation.

$a = 1$

Step 7

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$a = - 1, a = 1$

Step 8