// Numbas version: exam_results_page_options {"name": "Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "

Apply the quadratic formula to find the roots of a given equation. The quadratic formula is given in the steps if the student requires it.

", "licence": "Creative Commons Attribution 4.0 International"}, "extensions": [], "type": "question", "ungrouped_variables": ["a1", "a2", "a3", "a4", "b1", "b2", "b3", "b4", "x1", "p1", "p2", "x2", "a", "m"], "rulesets": {}, "advice": "

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\$x={\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\$

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a)

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From the equation, we can read off values for $a$, $b$ and $c$:

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\\\begin{align} a&=1\\text{,}\\\\ b&=\\var{a+m}\\text{,}\\\\ c&=\\var{a*m} \\text{.} \\end{align}\

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Substituting these values into the quadratic formula,

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\$x = \\frac {-\\var{a+m}\\pm\\sqrt{\\var{a+m}^2-4\\times \\var{a*m}}}{2}\\text{.}\$

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Note the $\\pm$ symbol in the formula. This means there are two solutions: one using $+$, the other using $-$.

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The two solutions are

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\\\begin{align} x_1&=\\var{m}\\text{,}\\\\ x_2&=\\var{a}\\text{.} \\end{align}\

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b)

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Note that the right-hand side of the given equation is not zero. We need to rewrite it in the form $ax^2+bx+c=0$:

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\\\begin{align} \\simplify{{a1}x^2+{a2}x+{a3}}&=\\var{a4}\\\\ \\simplify{{a1}x^2+{a2}x+{a3-a4}}&=0\\text{.} \\end{align}\

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Then we can read off values for $a$, $b$ and $c$:

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\\\begin{align} a&=\\var{a1}\\\\ b&=\\var{a2}\\\\ c&=\\var{a3-a4} \\text{.} \\end{align}\

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We can now substitute these values into the quadratic formula:

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\$x = {\\frac {-\\var{a2}\\pm\\sqrt{\\var{a2}^2-4\\times \\var{a1}\\times \\var{a3-a4}}}{2\\times\\var{a1}}}\\text{.}\$

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So the two solutions are

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\\\begin{align} x_1&=\\var{dpformat(x1,2)}\\\\ x_2&=\\var{dpformat(x2,2)}\\text{.} \\end{align}\

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c)

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We first rearrange our equation into the form $ax^2+bx+c=0$:

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\\\begin{align} \\simplify{{b1}x^2+{b2}x+{b3}}&=0=\\var{b4}x\\\\ \\simplify{{b1}x^2+{b2-b4}x+{b3}}&=0\\text{.} \\end{align}\

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We can then read off the values for $a, b$ and $c$, which are

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\\\begin{align} a&=\\var{b1}\\text{,}\\\\ b&=\\var{b2-b4}\\text{,}\\\\ c&=\\var{b3}\\text{.} \\end{align}\

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Substituting these values into the quadratic formula,

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\$x = {\\frac {-\\var{b2-b4}\\pm\\sqrt{\\var{b2-b4}^2-4\\times \\var{b1}\\times \\var{b3}}}{2\\times\\var{b1}}},\$

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we obtain solutions

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\\\begin{align} x_1&=\\var{dpformat(p1,2)}\\text{,}\\\\ x_2&=\\var{dpformat(p2,2)}\\text{.} \\end{align}\

", "variable_groups": [{"name": "part 2", "variables": ["b", "c"]}], "statement": "

When quadratic equations can't be factorised, or if equations are difficult to factorise (perhaps if the coefficients are large), we need to use the quadratic formula to solve the equations.

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Use the quadratic formula to calculate values for $x$ in these equations. Input the possible values as $x_1$ and $x_2$, where $x_1<x_2$.

", "name": "Using the Quadratic Formula to Solve Equations of the Form $ax^2 +bx+c=0$", "parts": [{"scripts": {}, "variableReplacements": [], "type": "gapfill", "prompt": "

$\\simplify{x^2+{a+m}x+{a*m}=0}$

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$x_1=$ [[0]]

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$x_2=$ [[1]]

", "stepsPenalty": 0, "steps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "information", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "

An equation of the form

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\$ax^2+bx+c=0\\text{,}\$

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can be solved using the quadratic formula

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\$x={\\frac {-b\\pm\\sqrt{b^2-4\\times a\\times c}}{2a}}\\text{.}\$

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$\\simplify{{a1}x^2+{a2}x+{a3}={a4}}$

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$x_1=$ [[0]]

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$x_2=$ [[1]]

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$\\simplify{{b1}x^2+{b2}x+{b3}={b4}x}$
$x_1=$ [[0]]
$x_2=$ [[1]]