Algebra Examples

$\frac{8}{{(x - 2)}^{2}} = \frac{4}{x^{2} - 9} + \frac{2}{3}$

Step 1

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

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Step 1.1

Subtract $\frac{4}{x^{2} - 9}$ from both sides of the equation.

$\frac{8}{{(x - 2)}^{2}} - \frac{4}{x^{2} - 9} = \frac{2}{3}$

Step 1.2

Subtract $\frac{2}{3}$ from both sides of the equation.

$\frac{8}{{(x - 2)}^{2}} - \frac{4}{x^{2} - 9} - \frac{2}{3} = 0$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{8}{{(x - 2)}^{2}} - \frac{4}{x^{2} - 3^{2}} - \frac{2}{3} = 0$

Step 2.2

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 $b = 3$ .

$\frac{8}{{(x - 2)}^{2}} - \frac{4}{(x + 3) (x - 3)} - \frac{2}{3} = 0$

Step 3

Set the denominator in $\frac{8}{{(x - 2)}^{2}}$ equal to $0$ to find where the expression is undefined.

${(x - 2)}^{2} = 0$

Step 4

Solve for $x$ .

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Step 4.1

Set the $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5

Set the denominator in $\frac{4}{(x + 3) (x - 3)}$ equal to $0$ to find where the expression is undefined.

$(x + 3) (x - 3) = 0$

Step 6

Solve for $x$ .

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Step 6.1

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

$x + 3 = 0$

$x - 3 = 0$

Step 6.2

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 6.2.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 6.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6.3

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 6.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 6.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 6.4

The final solution is all the values that make $(x + 3) (x - 3) = 0$ true.

$x = - 3, 3$

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 $0$ .

$x = - 3, x = 2, x = 3$

Step 8