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Question

Factorize each of the following algebraic expression: x2 − 11x − 42

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Solution

$\mathrm{To}\mathrm{factorise}{\mathrm{x}}^{2}-11\mathrm{x}-42,\mathrm{we}\mathrm{will}\mathrm{find}\mathrm{two}\mathrm{numbers}\mathrm{p}\mathrm{and}\mathrm{q}\mathrm{such}\mathrm{that}\mathrm{p}+\mathrm{q}=-11\mathrm{and}\mathrm{pq}=-42.\phantom{\rule{0ex}{0ex}}\mathrm{Now},\phantom{\rule{0ex}{0ex}}3+\left(-14\right)=-22\phantom{\rule{0ex}{0ex}}\mathrm{and}\phantom{\rule{0ex}{0ex}}3×\left(-14\right)=42\phantom{\rule{0ex}{0ex}}\mathrm{Splitting}\mathrm{the}\mathrm{middle}\mathrm{term}-11\mathrm{x}\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{quadratic}\mathrm{as}-14\mathrm{x}+3\mathrm{x},\mathrm{we}\mathrm{get}:\phantom{\rule{0ex}{0ex}}{\mathrm{x}}^{2}-11\mathrm{x}-42={\mathrm{x}}^{2}-14\mathrm{x}+3\mathrm{x}-42\phantom{\rule{0ex}{0ex}}=\left({\mathrm{x}}^{2}-14\mathrm{x}\right)+\left(3\mathrm{x}-42\right)\phantom{\rule{0ex}{0ex}}=\mathrm{x}\left(\mathrm{x}-14\right)+3\left(\mathrm{x}-14\right)\phantom{\rule{0ex}{0ex}}=\left(\mathrm{x}+3\right)\left(\mathrm{x}-14\right)$

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