Describe and correct the error in finding the product: $(k + 4)^2 = k^2 + 4^2 = k^2 + 16$
Understand the Problem
The question asks us to identify and correct the error in the expansion of the binomial expression $(k+4)^2$. The provided solution incorrectly squares each term individually, instead of using the correct binomial expansion formula or the FOIL method.
Answer
$(k+4)^2 = k^2 + 8k + 16$
Answer for screen readers
The error was squaring each term individually. The correct expansion is: $(k+4)^2 = k^2 + 8k + 16$
Steps to Solve
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Identify the error The error is in how the binomial $(k+4)$ is squared. The student squared each term individually instead of using the correct formula for expanding $(a+b)^2$, which is $a^2 + 2ab + b^2$.
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Correct the expansion using the binomial formula Using the formula $(a+b)^2 = a^2 + 2ab + b^2$, where $a = k$ and $b = 4$, we have: $$ (k+4)^2 = k^2 + 2(k)(4) + 4^2 $$
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Simplify the expression Simplify the middle term and the last term: $$ k^2 + 8k + 16 $$
The error was squaring each term individually. The correct expansion is: $(k+4)^2 = k^2 + 8k + 16$
More Information
The binomial expansion formula is super helpful! It's derived directly expanding $(a+b)(a+b)$ using the distributive property (also known as FOIL).
Tips
A common mistake is to forget the middle term when squaring a binomial, i.e., thinking $(a+b)^2 = a^2 + b^2$. This ignores the $2ab$ term, which comes from multiplying $(a+b)(a+b)$ out fully. Always remember the middle term!
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