Find all values \(a\) for which there exists an ordered pair \((a,b)\) satisfying the following system of equations: \(\begin{align*} a + ab^2 & = 40b, \\ a - ab^2 & = -32b. \end{align*} \)
List only the values for \(a\).
\(a+ab^2 = 40b\\ a-ab^2 = -32b\\\)
\(\text{add them}\\ 2a=8b\\ b=\dfrac{a}{4}\)
\(\text{subtract them}\\ 2ab^2 = 72b\\ \text{if }b\neq 0\\ 2ab=72\\ 2\dfrac{a^2}{4} = 72\\ a^2 = 144\\ a=\pm 12\)
\(\text{and of course }a=b=0 \text{ is also a solution}\\ \text{so }a = 0, \pm 12\)
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