Let $\alpha , \beta \in \mathrm { R }$. If $\alpha , \beta ^ { 2 }$ are the roots of quadratic equation $x ^ { 2 } - p x + 1 = 0$ and $\alpha ^ { 2 } , \beta$ equation $x ^ { 2 } - q x + 8 = 0$, then the value r if $\frac { r } { 8 }$ is the arithmetic means of p and q, is

Since alpha and beta squared are the roots of x squared minus px plus 1 equals 0, their product gives alpha multiplied by beta squared equals 1. Also, alpha squared and beta are the roots of x squared minus qx plus 8 equals 0, so their product gives alpha squared multiplied by beta equals 8. Multiplying these two equations yields alpha cubed multiplied by beta cubed equals 8, meaning alpha multiplied by beta equals 2; substituting this back gives alpha equals 1 by 2 and beta equals 4. Using the sum of roots, p equals alpha plus beta squared, which is 1 by 2 plus 16, equaling 33 by 2, and q equals alpha squared plus beta, which is 1 by 4 plus 4, equaling 17 by 4. The arithmetic mean of p and q is (33 by 2 plus 17 by 4) divided by 2, which equals 83 by 16; setting this equal to r divided by 8 gives r equal to 83 by 2.

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