Set the denominator in <math><mstyle displaystyle="true"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>3</mn></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>27</mn></mrow></mfrac></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to find where the expression is undefined.

Factor <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>27</mn></mstyle></math> using the AC method.

Consider the form <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mstyle></math> . Find a pair of integers whose product is <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> . In this case, whose product is <math><mstyle displaystyle="true"><mn>27</mn></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mo>-</mo><mn>12</mn></mstyle></math> .

Write the factored form using these integers.

If any individual factor on the left side of the equation is equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , the entire expression will be equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Set the first factor equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and solve.

Set the first factor equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> to both sides of the equation.

Set the next factor equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and solve.

Set the next factor equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> to both sides of the equation.

The final solution is all the values that make <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>9</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mstyle></math> true.

The equation is undefined where the denominator equals <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , the argument of a square root is less than <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , or the argument of a logarithm is less than or equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

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