If both \((x - 2)\) and \(\left(x - \frac{1}{2} \right)\) are the factors of \(ax^2 + 5x + b\), then show that \(a = b\).
Proof:
Let \(p(x) = ax^2 + 5x + b\)
By factor theorem, \((x - 2)\) and \(\left(x - \frac{1}{2} \right)\) are the factors of \(p(x)\), if \(p(2) = 0\) and \(p\left(\frac{1}{2} \right) = 0\)
\(p(2) =\) ..............(1)
\(p\left(\frac{1}{2} \right) =\) ..............(2)
Equating equations (\(1\)) and (\(2\)), we get:
\(4a + b = \)
\(3a = \)
\(a \)\( b\)
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