If left|begin{array}{ccc}9 & 25 & 16 16 & 36 | vedclass

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(D) Let the determinant be $D = \left|\begin{array}{ccc}9 & 25 & 16 \ 16 & 36 & 25 \ 25 & 49 & 36\end{array}ight|$.Applying the column operations

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$x^2-17 x+72=0$

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(D) Let the determinant be $D = \left|\begin{array}{ccc}9 & 25 & 16 \ 16 & 36 & 25 \ 25 & 49 & 36\end{array}ight|$.Applying the column operations $C_2 	o C_2 - C_1$ and $C_3 	o C_3 - C_2$:$D = \left|\begin{array}{ccc}9 & 16 & -9 \ 16 & 20 & -11 \ 25 & 24 & -13\end{array}ight|$.Alternatively, observe the pattern $a_n = (n+2)^2$. The determinant is $\left|\begin{array}{ccc}3^2 & 5^2 & 4^2 \ 4^2 & 6^2 & 5^2 \ 5^2 & 7^2 & 6^2\end{array}ight|$.Using row operations $R_2 	o R_2 - R_1$ and $R_3 	o R_3 - R_2$:$R_2 - R_1 = (16-9, 36-25, 25-16) = (7, 11, 9)$.$R_3 - R_2 = (25-16, 49-36, 36-25) = (9, 13, 11)$.Now, $D = \left|\begin{array}{ccc}9 & 25 & 16 \ 7 & 11 & 9 \ 9 & 13 & 11\end{array}ight|$.Applying $R_3 	o R_3 - R_2$:$D = \left|\begin{array}{ccc}9 & 25 & 16 \ 7 & 11 & 9 \ 2 & 2 & 2\end{array}ight| = 2 \left|\begin{array}{ccc}9 & 25 & 16 \ 7 & 11 & 9 \ 1 & 1 & 1\end{array}ight|$.Expanding along $R_3$: $D = 2 [1(225-176) - 1(81-112) + 1(99-175)] = 2 [49 + 31 - 76] = 2 [4] = 8$.Thus, $K = 8$. The roots are $K=8$ and $K+1=9$.The quadratic equation with roots $8$ and $9$ is $(x-8)(x-9) = x^2 - 17x + 72 = 0$.

If $\left|\begin{array}{ccc}9 & 25 & 16 \\ 16 & 36 & 25 \\ 25 & 49 & 36\end{array}\right|=K$, then $K, K+1$ are the roots of the equation

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