This question tests the students ability to factorise simple quadratic equations (where the coefficient of the x^2 term is 1) and use the factorised equation to solve the equation when it is equal to 0.
"}, "name": "Ida's copy of Solving Quadratic Equations with $x^2$ Coefficients of 1 by Factorising", "variable_groups": [{"variables": ["v6", "v5", "v4", "v3", "v2", "v1", "a", "b"], "name": "Part A "}, {"variables": ["r2", "r1"], "name": "Part B"}, {"variables": ["b1", "b2", "b3", "b4"], "name": "last q"}], "advice": "When we expand a factorised quadratic expression we obtain
\n\\[(x+a)(x+b)=x^2+(a+b)x+ab\\text{.}\\]
\nThis means when factorising the quadratic expression
\n\\[\\simplify{x^2+{r1+r2}x+{r1*r2}=0}\\]
\nwe need to find two values that add together to make $\\var{r1+r2}$ and multiply together to make $\\var{r1*r2}$.
\n\\[\\begin{align}
\\var{r1} \\times \\var{r2}&=\\var{r1*r2}\\\\
\\var{r1}+\\var{r2}&=\\var{r1+r2}\\\\
\\end{align} \\]
Therefore using $\\var{r1}$ and $\\var{r2}$ we can write out the factorised equation
\n\\[\\simplify{(x+{r1})(x+{r2})}=0\\text{.}\\]
\n\nIn order to find the values of $x$ we need to factorise the questions like in part a). To do this we need to find factors of $\\var{r1*r2}$ that add together to give $\\var{r1+r2}$, which could be $\\var{r1}$ and $\\var{r2}$.
\nThis would give us a factorised equation of
\n\\[\\simplify {(x+{r1})(x+{r2})}=0\\text{.}\\]
\nIn order to solve for $x$ we need $x$ values that would mean that
\n\\[\\begin{align}
(\\simplify {(x+{r1})})&=0 \\\\
\\text{or}&\\\\
(\\simplify{(x+{r2})})&=0\\text{.}
\\end{align}\\]
If at least one of the factorised brackets equals $0$ then our equation is satisfied because $0\\times(x+a)=0$.
\nTherefore our possible $x$ values are
\n\\[\\begin{align}
x_1&=\\var{-r2}\\\\
x_2&=\\var{-r1}\\text{.}
\\end{align}\\]
Løs andregradslikningen $\\simplify{x^2 +{r1+r2}x+{r1*r2}}=0\\text{.}$
\nInput your values as $x_1$ and $x_2$, where $x_1<x_2$. skriv detta tydligt!
\n$x_1=$ [[0]]
\n$x_2=$ [[1]]
", "marks": 0, "gaps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "maxValue": "-r2", "minValue": "-r2", "marks": 1, "mustBeReduced": false, "allowFractions": false, "scripts": {}, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "type": "numberentry", "mustBeReducedPC": 0}, {"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "maxValue": "-r1", "minValue": "-r1", "marks": 0, "mustBeReduced": false, "allowFractions": false, "scripts": {}, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "type": "numberentry", "mustBeReducedPC": 0}], "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst"}, {"showCorrectAnswer": true, "variableReplacements": [], "type": "gapfill", "prompt": "Faktoriser andregradslikningen $\\simplify{x^2 +{r1+r2}x+{r1*r2}}=0\\text{.}$
\n[[0]] $=0$
", "marks": 0, "gaps": [{"checkingaccuracy": 0.001, "checkvariablenames": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "vsetrangepoints": 5, "marks": 1, "notallowed": {"message": "", "showStrings": false, "strings": ["^"], "partialCredit": 0}, "showFeedbackIcon": true, "scripts": {}, "expectedvariablenames": [], "showpreview": true, "showCorrectAnswer": true, "checkingtype": "absdiff", "answer": "(x+{r2})(x+{r1})", "musthave": {"message": "", "showStrings": false, "strings": ["(", ")"], "partialCredit": 0}, "vsetrange": [0, 1], "type": "jme"}], "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst"}, {"showCorrectAnswer": true, "variableReplacements": [], "type": "gapfill", "prompt": "Løs ulikheten $\\simplify{x^2 +{r1+r2}x+{r1*r2}}>0\\text{.}$
\nSvar:
\n$x>$ [[0]]
\n[[2]]
\n$x<$ [[1]]
", "marks": 0, "gaps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "maxValue": "-r1", "minValue": "-r1", "marks": 1, "mustBeReduced": false, "allowFractions": false, "scripts": {}, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "type": "numberentry", "mustBeReducedPC": 0}, {"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "maxValue": "-r2", "minValue": "-r2", "marks": 1, "mustBeReduced": false, "allowFractions": false, "scripts": {}, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "type": "numberentry", "mustBeReducedPC": 0}, {"matrix": [0, "1"], "minMarks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "choices": ["Og
", "eller
"], "distractors": ["", ""], "marks": 0, "displayType": "radiogroup", "scripts": {}, "showFeedbackIcon": true, "shuffleChoices": false, "showCorrectAnswer": true, "displayColumns": 0, "maxMarks": 0, "type": "1_n_2"}], "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst"}], "statement": "Andregradslikningen $ax^2+bx+c=0$ har løsningene
\n\\[x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a},\\space \\text{når}\\space b^2-4ac\\geq0\\text{.}\\]
\nDersom andregradsuttrykket $ax^2+bx+c$ har nullpunktene $x=x_{1}$ og $x=x_{2}$ kan det faktoriseres til $a(x-x_{1})(x-x_{2}).$