A binary operation * is defined by a * b = a\(^b\). If a * 2 = 2 - a, find the possible values of a.
The correct answer is D. 1, -2
Given that \(a * 2 = 2 - a\), let's substitute \(2\) for \(b\) in the definition of the binary operation:
\[a * 2 = a^2\]
Now we have the equation: \(a^2 = 2 - a\).
Rearrange the equation to a quadratic form:
\[a^2 + a - 2 = 0\]
Now, let's factor the quadratic equation:
\[(a + 2)(a - 1) = 0\]
Setting each factor to zero and solving for \(a\):
\(a + 2 = 0 \implies a = -2\)
\(a - 1 = 0 \implies a = 1\)
So, the possible values of \(a\) are \(1\) and \(-2\).
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