Chapter: Uncategorized· Mathematics

Given that x = -9 is a root of ||The other two roots are and __ .

Given that x = -9 is a root of \(\begin{array}{|ccc|cc|} x&3&7\\2&x&2\\7 &6&x\\ \end{array}=0\) The other two roots are ______ and ________ .

Doubtly Asked 11 Apr 2021 Updated 23 Oct 2026 Student doubt jeemain cengage Save WhatsApp Report

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Doubtly · 11 Apr 2021 Report

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Answer: The other two roots are 2 and 7.

Expand the determinant along the first row:

\[\begin{vmatrix} x & 3 & 7\\ 2 & x & 2\\ 7 & 6 & x \end{vmatrix} = x(x^2-12) - 3(2x-14) + 7(12-7x)\]

\[= x^3 - 12x - 6x + 42 + 84 - 49x\]

\[= x^3 - 67x + 126\]

Check \(x=-9\): \(-729 + 603 + 126 = 0\). So \((x+9)\) is a factor. Dividing,

\[x^3 - 67x + 126 = (x+9)(x^2 - 9x + 14)\]

\[x^2 - 9x + 14 = (x-2)(x-7)\]

So the roots are \(-9, 2, 7\); the other two roots are 2 and 7.

Quick check: the sum of roots of \(x^3 + 0x^2 - 67x + 126 = 0\) is 0, and \(-9 + 2 + 7 = 0\).

Doubtly AI · 23 Oct 2026 AI-verified · Oct 2026 Report

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