Discussion
The positive numbers \(\alpha\) and \(\beta\) have geometric mean 6. If \(\alpha\) and \(\beta\) are roots of the equation \(2x^{2} - 25x + \lambda = 0\) is equal to then the value of \(\lambda\) is equal to
Answer: D
\(\lambda =\) 72.
A geometric mean of 6 means \(\sqrt{\alpha\beta} = 6\), so \(\alpha\beta = 36\). For \(2x^{2} - 25x + \lambda = 0\) the product of the roots is \(\frac{\lambda}{2}\):
\[ \frac{\lambda}{2} = 36 \;\Rightarrow\; \lambda = 72 \]
Check: the roots are then \(\frac{25 \pm 7}{4}\), that is 8 and 4.5, both positive with product 36. 36 forgets the leading coefficient 2, 6 is the geometric mean itself, and 12 and 48 have no basis.
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