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  • Fill in the blanks If x = -9 is a root of , then other two roots are ___.

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If x = -9 is a root of \left|\begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array}\right|=0 , then other two roots are ___.

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\\ \begin{aligned} &\text { We know that }\\ &x=-9 \text { is a root of }\left|\begin{array}{lll} x & 3 & 7 \\ 2 & x & 2 \\ 7 & 6 & x \end{array}\right|=0\\ &\Rightarrow \mathrm{x}\left|\begin{array}{ll} \mathrm{x} & 2 \\ 6 & \mathrm{x} \end{array}\right|-3\left|\begin{array}{ll} 2 & 2 \\ 7 & \mathrm{x} \end{array}\right|+7\left|\begin{array}{ll} 2 & \mathrm{x} \\ 7 & 6 \end{array}\right|=0 \end{aligned}

\\ \Rightarrow x [x^{2} - 12] - 3 [2x - 14] + 7 [12 - 7x] = 0\\ \\ \Rightarrow x^{3} - 12x - 6x + 42 + 84 - 49x = 0\\ \\ \Rightarrow x^{3} - 67x + 126 = 0\\ (x + 9)(2 - x)(7 - x) = 0 \\ Here, 126 \times 1 = 9 \times 2 \times 7\\ \\ For x = 2,\\ \\ \Rightarrow 2^{3} - 67 \cdot 2 + 126 = 134 - 134 = 0\\ \\ \therefore x = 2 \text{ is one root}.\\ \\ For x = 7,\\ \\ \Rightarrow 7^{3} - 67 \cdot 7 + 126 = 469 - 469 = 0\\ \\ \therefore x = 7 \text{ will be another root}.\\

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