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{"name": "AD3 Completing the square", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "AD3 Completing the square", "tags": ["complete the square", "completing the square", "taxonomy"], "metadata": {"description": "<p>Rearrange expressions in the form $ax^2+bx+c$ to $a(x+b)^2+c$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a&nbsp;square plus another term - this is called \"completing the square\".</p>\n<p>This can be useful when it isn't obvious how to fully factorise a quadratic equation.</p>\n<p>Rewrite the following expressions in the form \\[(x+b)^2-c\\]</p>", "advice": "<p>Completing the square works by noticing that</p>\n<p>\\[ (x+a)^2 = x^2 + 2ax + a^2 \\]</p>\n<p>So when we see an expression of the form $x^2 + 2ax$, we can rewrite it as $(x+a)^2-a^2$.</p>\n<p></p>\n<p>Replace $x^2+\\var{evens2}x$ with $(x+\\var{evens2/2})^2 - \\var{evens2/2}^2$. Remember&nbsp;to keep the $\\var{evens2-evens1}$ term on the end!</p>\n<p>\\begin{align}<br/>\\simplify[basic]{ x^2 + {evens2}x + {evens2-evens1}} &nbsp;&amp;= \\simplify[basic]{ (x+{evens2/2})^2 - {evens2/2}^2 + {evens2-evens1} } \\\\<br/>&amp;= \\simplify[basic]{ (x+{evens2/2})^2 + {evens2-evens1 - evens2^2/4} }<br/>\\end{align}</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Quadratics\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></p>\n<h4><em></em></h4>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"multiall2": {"name": "multiall2", "group": "Ungrouped variables", "definition": "all2*random(2..10 #2)", "description": "", "templateType": "anything", "can_override": false}, "odds3": {"name": "odds3", "group": "Odds and Evens", "definition": "random(11..30 #2 except odds odds2)", "description": "", 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