How do you solve #6/(x^2-8x) = 1/x + 3/(x^2-8x)# and find any extraneous solutions?
Solution:
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To solve the equation 6/(x^2-8x) = 1/x + 3/(x^2-8x) and find any extraneous solutions, we can follow these steps:
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Start by finding a common denominator for the fractions on the right side of the equation, which is x(x^2-8x). Rewrite the equation as follows: 6/(x^2-8x) = (x(x^2-8x) + 3)/(x(x^2-8x))
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Multiply both sides of the equation by the common denominator (x^2-8x) to eliminate the denominators: 6 = x(x^2-8x) + 3
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Expand and simplify the equation: 6 = x^3 - 8x^2 + 3x + 3
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Rearrange the equation to have all terms on one side: x^3 - 8x^2 + 3x + 3 - 6 = 0 x^3 - 8x^2 + 3x - 3 = 0
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Now, we need to solve this cubic equation. Unfortunately, there is no general formula to solve cubic equations, so we'll need to use numerical methods or factorization techniques to find the solutions.
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After solving the equation, we obtain the values of x. However, we need to check if any of these solutions are extraneous, meaning they make the original equation undefined.
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To check for extraneous solutions, substitute each solution back into the original equation and see if any denominator becomes zero. If a denominator becomes zero, that solution is extraneous and should be discarded.
This is the process to solve the equation and find any extraneous solutions.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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