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Quiz covering basic arithmetic with complex numbers and solving roots for a quadratic with complex solutions

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Given the complex numbers

\\[a=\\var{z1}\\\\b=\\var{z2}\\\\c=\\var{z3}\\\\d=\\var{z4}\\]

calculate the following. Enter your answers in the form

\\[x+yi\\]

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$a+b=$

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No brackets are required or allowed.

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$a+c=$

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No brackets are required or allowed.

$a-b=$

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No brackets are required or allowed.

$a-c=$

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No brackets are required or allowed.

$a+d=$

", "answer": "{z1+z4}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

No brackets are required or allowed.

$\\simplify{{n}*b+{m}*d}=$

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This question provides practice at adding, subtracting, dividing and multiplying complex numbers in rectangular form.

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Given the complex numbers

\\[a=\\var{a},\\quad b=\\var{b},\\quad c=\\var{c},\\quad d=\\var{d},\\]

calculate the following. Enter your answers in the form

\\[x+iy\\]

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$ab=$ [[0]]

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Expand the brackets $(\\var{a})(\\var{b})$.

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Simplify using the fact $i^2=-1$.

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No brackets are required or allowed.

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$ac=$ [[0]]

No brackets are required or allowed.

$\\frac{a}{b}=$ [[0]]

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What is the complex conjugate of $b$?

$\\overline{b}=$

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Multiply the numerator ($\\var{a}$) by $\\overline{b}$.

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Multiply the denominator ($\\var{b}$) by $\\overline{b}$.

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Now divide your two answers.

\\[\\frac{a}{b}=\\frac{a\\overline{b}}{b\\overline{b}}\\]

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No brackets are required or allowed.

$\\frac{a}{c}=$ [[0]]

No brackets are required or allowed.

$ad=$ [[0]]

No brackets are required or allowed.

$\\frac{b}{d}=$ [[0]]

No brackets are required or allowed.

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Solve the quadratic equation:

\\[\\simplify{x^2-2*{re(a)}*x+{abs(a)^2}}=0\\]

You may use the following formula: if $ax^2+bx+c=0$ then

\\[x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}\\]

Enter your answers in the form $x+yi$.

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$x_1=\\;$ [[0]] (solution with positive imaginary part)

$x_2=\\;$ [[1]] (solution with negative imaginary part)

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