// Numbas version: exam_results_page_options {"name": "Polynomials: Expand and Simplify 1a", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Polynomials: Expand and Simplify 1a", "tags": ["Category: Working with polynomials"], "metadata": {"description": "

Finding linear combinations of two quadtratic expressions of the form $ax^2+bx+c$.

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Given two functions, \\[ f(x) = \\simplify{{a1}x^2+{b1}x+{c1}}, \\quad g(x)=\\simplify{{a2}x^2+{b2}x+{c2}},\\]

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calculate \\[\\simplify{{m}f(x)+{n}g(x)}.\\]

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Substituting the functions  $f(x) = \\simplify{{a1}x^2+{b1}x+{c1}}$ and $g(x)=\\simplify{{a2}x^2+{b2}x+{c2}}$ into $\\simplify{{m}f(x)+{n}g(x)}$:

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\\[ \\simplify{{m}f(x)+{n}g(x)} = \\simplify{{m}({a1}x^2+{b1}x+{c1})+{n}({a2}x^2+{b2}x+{c2})}. \\]

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Multiplying out the brackets and collecting similar terms:

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\\[ \\begin{split}\\simplify{{m}({a1}x^2+{b1}x+{c1})+{n}({a2}x^2+{b2}x+{c2})}&\\,=\\simplify[all, !cancelTerms,!collectNumbers]{{m*a1}x^2+{m*b1}x+{m*c1}+{n*a2}x^2+{n*b2}x+{n*c2}} \\\\ &\\,=\\simplify[all, !noLeadingminus]{{m*a1}x^2+{m*b1}x+{m*c1}+{n*a2}x^2+{n*b2}x+{n*c2}}\\end{split} \\]

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