Algebra Examples

Find the Holes in the Graph (3x^2-108)/(x^2+2x-24)
Step 1
Factor .
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Step 1.1
Factor out of .
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Step 1.1.1
Factor out of .
Step 1.1.2
Factor out of .
Step 1.1.3
Factor out of .
Step 1.2
Rewrite as .
Step 1.3
Factor.
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Step 1.3.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.3.2
Remove unnecessary parentheses.
Step 2
Factor using the AC method.
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Step 2.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 2.2
Write the factored form using these integers.
Step 3
Cancel the common factor of .
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Step 3.1
Cancel the common factor.
Step 3.2
Rewrite the expression.
Step 4
To find the holes in the graph, look at the denominator factors that were cancelled.
Step 5
To find the coordinates of the holes, set each factor that was cancelled equal to , solve, and substitute back in to .
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Step 5.1
Set equal to .
Step 5.2
Subtract from both sides of the equation.
Step 5.3
Substitute for in and simplify.
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Step 5.3.1
Substitute for to find the coordinate of the hole.
Step 5.3.2
Simplify.
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Step 5.3.2.1
Cancel the common factor of and .
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Step 5.3.2.1.1
Factor out of .
Step 5.3.2.1.2
Cancel the common factors.
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Step 5.3.2.1.2.1
Factor out of .
Step 5.3.2.1.2.2
Factor out of .
Step 5.3.2.1.2.3
Factor out of .
Step 5.3.2.1.2.4
Cancel the common factor.
Step 5.3.2.1.2.5
Rewrite the expression.
Step 5.3.2.2
Subtract from .
Step 5.3.2.3
Subtract from .
Step 5.3.2.4
Multiply by .
Step 5.3.2.5
Dividing two negative values results in a positive value.
Step 5.4
The holes in the graph are the points where any of the cancelled factors are equal to .
Step 6