Let $\displaystyle \alpha_1$ and $\displaystyle \alpha_2$ be the roots of the equation $\displaystyle x^2 - 4x + P_1 = 0$, and $\displaystyle \alpha_3$ and $\displaystyle \alpha_4$ be the roots of the equation $\displaystyle x^2 - 36x + P_2 = 0$. If $\displaystyle \alpha_1 < \alpha_2 < \alpha_3 < \alpha_4$ and $\displaystyle \alpha_1, \alpha_2, \alpha_3, \alpha_4$ are in G.P., then the product $\displaystyle P_1P_2$ equals

Let the common ratio of the geometric progression be r. The sums of the roots give alpha 1 plus alpha 2 equals 4, so we can write a plus ar equals 4, where a is alpha 1; similarly, alpha 3 plus alpha 4 equals 36 gives ar squared plus ar cubed equals 36. Factoring out r squared from the second equation yields r squared times (a plus ar) equals 36, and substituting the first equation gives 4r squared equals 36, so r equals 3. Since a times (1 plus r) equals 4, a equals 1, making alpha 1 equals 1 and alpha 2 equals 3; then alpha 3 equals 9 and alpha 4 equals 27, so P1 equals 1 times 3 and P2 equals 9 times 27. Multiplying P1 and P2 gives 3 times 243, which equals 729. The result is 729.

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