1955 AHSME Problems/Problem 47

Problem 47

The expressions $a+bc$ and $(a+b)(a+c)$ are:

$\textbf{(A)}\ \text{always equal}\qquad\textbf{(B)}\ \text{never equal}\qquad\textbf{(C)}\ \text{equal whenever }a+b+c=1\\ \textbf{(D)}\ \text{equal when }a+b+c=0\qquad\textbf{(E)}\ \text{equal only when }a=b=c=0$

Solution

Using the FOIL method, we see that $(a+b)(a+c) = a^2 + ab + ac + bc.$ We want to solve $a + bc = a^2 + ab + ac + bc$ $a = a^2 + ab + ac$ $a((a + b + c) - 1) = 0$ These expressions are equal only when $\boxed{\text{(C) equal whenever } a + b + c = 1.}$

1955 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 46	Followed by Problem 48
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1955 AHSME Problems/Problem 47

Problem 47

Solution

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