If the difference of the squares of the roots of the equation \(x^2 - 6x + q = 0\) is 24, then find the value of \(q\).
5
For the equation \(x^2 - 6x + q = 0\), let the roots be α and β.
Sum of roots α + β = 6 and product of roots αβ = q.
Difference of squares: |α2 − β2| = |α + β| · |α − β| = 6 · |α − β| = 24, so |α − β| = 4.
Use (α − β)2 = (α + β)2 − 4αβ: 16 = 36 − 4q, giving 4q = 20 and q = 5.
Hence, 5.
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