// Numbas version: finer_feedback_settings
{"name": "10e. Solving (x+a)/b=c/(x+d)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "10e. Solving (x+a)/b=c/(x+d)", "tags": [], "metadata": {"description": "<p>Solving equations involving algebraic fractions which simplify into quadratics</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve the following equation for $x$.</p>\n<p>\\[\\simplify{(x+{a})/{b}}=\\simplify{{c}/(x+{d})}\\]</p>", "advice": "<p>First multiply through by $\\var{b}$ and $\\simplify{x+{d}}$ and re-arrange to get&nbsp;</p>\n<p>\\[\\simplify{x^2+{xCoff}x+{Const}=0}\\]</p>\n<p>To solve a quadratic equation of the form \\[ x^2+bx+c=0\\] by factorisation, we want to factorise the equation into the form \\[(x+p)(x+q)=0,\\] where $p+q=b$ and $p \\times q = c$.&nbsp;</p>\n<p>Hence, for the equation \\[\\simplify{x^2+{xCoff}x+{Const}=0}, \\]</p>\n<p>this can be factorised to \\[\\simplify{(x+{x1})(x+{x2})=0}.\\] This equation is satisfied when either \\[\\simplify{x+{x1}=0} \\quad \\text{or} \\quad \\simplify{x+{x2}=0}, \\] which implies the solutions to this quadratic equation are \\[ \\simplify{x={-x1}} \\quad \\text{and} \\quad \\simplify{x={-x2}} .\\]</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 resources to help you revise how to solve quadratic equations.</a></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(-12..12)", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(-12..12)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(separate(-100..100,x->x|(Const-a*d))[0])", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", 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