Which of these is missing from this numerator?
$\frac{2 c}{c^2-2 c-1}+\frac{4 c}{c^2+3 c-4}=\frac{?}{(c-1)(c-1)(c+4)}$
$6 c^2-8 c-4$
$2 c^2+4 c+8 c$
$2 c+4 c$
Answer (1)
Factored the denominator c 2 + 3 c − 4 into ( c + 4 ) ( c − 1 ) .
Identified a likely typo in the original problem, assuming the first denominator was ( c − 1 ) 2 instead of c 2 − 2 c − 1 .
Found a common denominator of ( c − 1 ) 2 ( c + 4 ) .
Simplified the numerator to 6 c 2 + 4 c , noting that none of the provided options are 'missing' from it.
Concluded that the problem is likely flawed, and if the question was asking which of the options IS the numerator, then the answer would be none of the above.
Explanation
Problem Setup We are given the equation c 2 − 2 c − 1 2 c + c 2 + 3 c − 4 4 c = ( c − 1 ) ( c − 1 ) ( c + 4 ) ? and we need to find the missing numerator.
Factoring the Denominator First, let's factor the denominator c 2 + 3 c − 4 . We are looking for two numbers that multiply to -4 and add to 3. These numbers are 4 and -1. So, we can factor the denominator as ( c + 4 ) ( c − 1 ) .
c 2 − 2 c − 1 2 c + ( c + 4 ) ( c − 1 ) 4 c = ( c − 1 ) ( c − 1 ) ( c + 4 ) ?
Finding a Common Denominator Now, we need to find a common denominator for the left side of the equation. The desired denominator is ( c − 1 ) ( c − 1 ) ( c + 4 ) = ( c − 1 ) 2 ( c + 4 ) . To get this common denominator, we need to multiply the first fraction by ( c − 1 ) ( c + 4 ) ( c − 1 ) ( c + 4 ) and the second fraction by c 2 − 2 c − 1 c 2 − 2 c − 1 .
( c 2 − 2 c − 1 ) ( c − 1 ) ( c + 4 ) 2 c ( c − 1 ) ( c + 4 ) + ( c + 4 ) ( c − 1 ) ( c 2 − 2 c − 1 ) 4 c ( c 2 − 2 c − 1 ) = ( c − 1 ) ( c − 1 ) ( c + 4 ) ? However, we notice that the denominator on the left side is NOT ( c − 1 ) ( c − 1 ) ( c + 4 ) . There must be a typo in the problem. Let's assume that the first denominator is ( c − 1 ) 2 instead of c 2 − 2 c − 1 . Then the equation becomes ( c − 1 ) 2 2 c + ( c + 4 ) ( c − 1 ) 4 c = ( c − 1 ) ( c − 1 ) ( c + 4 ) ?
Rewriting with Common Denominator Now, we find a common denominator of ( c − 1 ) 2 ( c + 4 ) . We multiply the first fraction by c + 4 c + 4 and the second fraction by c − 1 c − 1 .
( c − 1 ) 2 ( c + 4 ) 2 c ( c + 4 ) + ( c + 4 ) ( c − 1 ) ( c − 1 ) 4 c ( c − 1 ) = ( c − 1 ) ( c − 1 ) ( c + 4 ) ? ( c − 1 ) 2 ( c + 4 ) 2 c ( c + 4 ) + 4 c ( c − 1 ) = ( c − 1 ) ( c − 1 ) ( c + 4 ) ?
Simplifying the Numerator Now, we simplify the numerator: 2 c ( c + 4 ) + 4 c ( c − 1 ) = 2 c 2 + 8 c + 4 c 2 − 4 c = 6 c 2 + 4 c So the equation becomes ( c − 1 ) 2 ( c + 4 ) 6 c 2 + 4 c = ( c − 1 ) ( c − 1 ) ( c + 4 ) ?
Comparing with Given Options Comparing the simplified numerator 6 c 2 + 4 c to the given options: 6 c 2 − 8 c − 4 2 c 2 + 4 c + 8 c = 2 c 2 + 12 c 2 c + 4 c = 6 c None of the options match our result. However, if the question was: Which of these is the numerator? Then the answer would be none of the above.
Assuming a Typo and Finding the Numerator Let's assume there was a typo in the original problem and the first denominator was ( c − 1 ) 2 and the question was asking for the numerator. Then the numerator is 6 c 2 + 4 c .
Considering Another Typo However, the question asks 'Which of these is missing from this numerator?' and the options are: 6 c 2 − 8 c − 4 2 c 2 + 4 c + 8 c 2 c + 4 c Since we found the numerator to be 6 c 2 + 4 c , none of the options are 'missing' from it in a direct additive sense. The problem is likely flawed. However, if we assume the first denominator was indeed c 2 − 2 c − 4 instead of c 2 − 2 c − 1 , then the common denominator would be ( c 2 − 2 c − 1 ) ( c + 4 ) ( c − 1 ) . Then the numerator would be 2 c ( c + 4 ) ( c − 1 ) + 4 c ( c 2 − 2 c − 1 ) = 2 c ( c 2 + 3 c − 4 ) + 4 c 3 − 8 c 2 − 4 c = 2 c 3 + 6 c 2 − 8 c + 4 c 3 − 8 c 2 − 4 c = 6 c 3 − 2 c 2 − 12 c . This is still not any of the options.
Final Answer with Typo Assumption Given the likely typo in the problem, and assuming the first denominator was ( c − 1 ) 2 , the numerator is 6 c 2 + 4 c . None of the options are 'missing' from this numerator. If the question was asking which of the options IS the numerator, then the answer would be none of the above.
Examples
When combining different sources of ingredients in a recipe, you often need to find a common denominator to understand the total quantity of a particular component. For instance, if one ingredient provides ( c − 1 ) 2 2 c amount of sugar and another provides ( c + 4 ) ( c − 1 ) 4 c amount, finding the common denominator helps determine the total sugar content accurately. This is crucial for maintaining the recipe's balance and flavor profile. Understanding how to manipulate and combine these fractions ensures the final dish has the intended sweetness and consistency.
