The coefficient of \(x^2\) in the expansion of \( (2-qx)^4-\left(1+\frac8q x\right)^6 \) is \(324\).
Find the possible values of the constant \(q\).
Solution
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The coefficient of \(x^2\) in \( (2-qx)^4 \) is
\( \binom42(2)^2(-q)^2=24q^2 \).
The coefficient of \(x^2\) in \( \left(1+\frac8q x\right)^6 \) is
\( \binom62\left(\frac8q\right)^2=\frac{960}{q^2} \).
Since the second expansion is subtracted,
\( 24q^2-\frac{960}{q^2}=324 \).
Multiply by \(q^2\):
\( 24q^4-324q^2-960=0 \).
Divide by \(12\):
\( 2q^4-27q^2-80=0 \).
Let \(u=q^2\). Then
\( 2u^2-27u-80=0 \).
\( (2u+5)(u-16)=0 \).
Since \(u=q^2\), \(u=16\).
Answer: \( q=4 \) or \( q=-4 \).
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