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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.

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#### a)

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When we expand a factorised quadratic expression we obtain

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\$(x+a)(x+b)=x^2+(a+b)x+ab\\text{.}\$

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This means when factorising the quadratic expression

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\$\\simplify{x^2+{r1+r2}x+{r1*r2}=0}\$

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we need to find two values that add together to make $\\var{r1+r2}$ and multiply together to make $\\var{r1*r2}$.

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\\\begin{align} \\var{r1} \\times \\var{r2}&=\\var{r1*r2}\\\\ \\var{r1}+\\var{r2}&=\\var{r1+r2}\\\\ \\end{align} \

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Therefore using $\\var{r1}$ and $\\var{r2}$ we can write out the factorised equation

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\$\\simplify{(x+{r1})(x+{r2})}=0\\text{.}\$

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#### b)

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In 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 $\\var{r2}$.

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This would give us a factorised equation of

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\$\\simplify {(x+{r1})(x+{r2})}=0\\text{.}\$

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In order to solve for $x$ we need $x$ values that would mean that

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\\\begin{align} (\\simplify {(x+{r1})})&=0 \\\\ \\text{or}&\\\\ (\\simplify{(x+{r2})})&=0\\text{.} \\end{align}\

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If at least one of the factorised brackets equals $0$ then our equation is satisfied because $0\\times(x+a)=0$.

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Therefore our possible $x$ values are

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\\\begin{align} x_1&=\\var{-r2}\\\\ x_2&=\\var{-r1}\\text{.} \\end{align}\

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Løs andregradslikningen $\\simplify{x^2 +{r1+r2}x+{r1*r2}}=0\\text{.}$

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Input your values as $x_1$ and $x_2$, where $x_1<x_2$. skriv detta tydligt!

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$x_1=$ [[0]]

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$x_2=$ [[1]]

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Faktoriser andregradslikningen $\\simplify{x^2 +{r1+r2}x+{r1*r2}}=0\\text{.}$

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[[0]] $=0$

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Løs ulikheten $\\simplify{x^2 +{r1+r2}x+{r1*r2}}>0\\text{.}$

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Svar:

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$x>$ [[0]]

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[[2]]

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$x<$ [[1]]

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Og

", "

eller

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Andregradslikningen $ax^2+bx+c=0$ har løsningene

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\$x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a},\\space \\text{når}\\space b^2-4ac\\geq0\\text{.}\$

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Dersom andregradsuttrykket $ax^2+bx+c$ har nullpunktene $x=x_{1}$ og $x=x_{2}$ kan det faktoriseres til  $a(x-x_{1})(x-x_{2}).$

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