Algebra Examples

$\frac{x^{2} - 9}{x^{2} - 6 x + 9} \cdot \frac{6 x - 18}{x + 3}$

Step 1

Set the denominator in $\frac{x^{2} - 9}{x^{2} - 6 x + 9}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 6 x + 9 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor using the perfect square rule.

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Step 2.1.1

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

$x^{2} - 6 x + 3^{2} = 0$

Step 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.

$6 x = 2 \cdot x \cdot 3$

Step 2.1.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 3 + 3^{2} = 0$

Step 2.1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

${(x - 3)}^{2} = 0$

Step 2.2

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

$x - 3 = 0$

Step 2.3

Add $3$ to both sides of the equation.

$x = 3$

Step 3

Set the denominator in $\frac{6 x - 18}{x + 3}$ equal to $0$ to find where the expression is undefined.

$x + 3 = 0$

Step 4

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 5

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 $0$ .

$x = - 3, x = 3$

Step 6