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    • ▾Algebra 2
      • ▸Polynomial arithmetic
        • •Descartes's rule of signs
        • •Difficult factoring problems
        • •Polynomial long division and graphing cubics
        • •Rational root theorem
        • •Higher-order polynomials
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        • •Reciprocal polynomials
        • •Adding and subtracting polynomials
        • •Multiplying binomials by polynomials
        • •Proving the sum and difference of cubes formulas
        • •Factoring and solving polynomials by graphing
      • ▾Complex numbers
        • •Intro to complex numbers
        • •Powers of the imaginary unit
        • •Simplifying square roots of negative integers
        • •Plotting complex numbers
        • •Adding and subtracting complex numbers
        • •Multiplying complex numbers
        • •Complex number conjugates
        • •Dividing complex numbers
        • •Modulus and argument of a complex number
        • •Proving properties of the complex modulus
        • •Converting complex numbers between polar and rectangular form
        • •Powers of complex numbers using modulus and argument
        • •Square root of a complex number
        • •Distance and midpoint of complex numbers
        • •Linear factorization
        • •Solving equations with complex numbers
        • •Quadratic equations with complex solutions
        • •Complex conjugate root theorem
        • •Fundamental theorem of algebra
      • ▸Polynomial factorization, division, and end behavior
        • •Finding the GCD and LCM of polynomials
        • •Sophie Germain's identity
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        • •Synthetic division
        • •Converting improper algebraic fractions to mixed algebraic fractions
      • ▸Rational exponents and radicals
        • •Adding and subtracting radicals
        • •Converting between radicals and rational exponents
        • •Rewrite exponential expressions
        • •Simplifying expressions with radicals and rational exponents
        • •Solving equations with radicals and rational exponents
        • •Transformations of the square and cube root functions
      • ▸Logarithms
        • •Evaluating logarithms
        • •Evaluating natural logarithms
        • •Converting between logarithmic form and exponential form
        • •Evaluating logarithmic expressions using the one-to-one property
        • ▸Expanding and condensing logarithmic expressions
          • •Expanding logarithmic expressions
          • ▸Condensing logarithmic expressions
            • •Lesson
            • •Practice
        • •Inverses of exponential and logarithmic functions
        • ▸Graphing logarithmic functions
          • •Sketching logarithmic functions under transformations
        • ▸Basic properties of logarithms
          • •Intro
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        • •Change of base formula
        • •Properties of logarithms cheat sheet
        • •Solving logarithmic inequalities
        • •Solving literal equations using properties of logarithms
        • ▸Solving logarithmic equations
          • •Solving logarithmic equations using the one-to-one property
          • •Solving logarithmic equations using the properties of logarithms
        • •Verifying logarithmic identities
        • •Fractal dimension
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      • ▸Transformations of functions
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          • ▸Solving exponential equations using the one-to-one property
            • •Lesson
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        • •Intersections of lines, circles, and parabolas
        • •Solving and graphing polynomial inequalities using a sign chart
        • •Solving and graphing rational inequalities using a sign chart
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      • ▸Rational functions
        • •Adding and subtracting rational expressions
        • •Finding the domain and range of rational functions
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        • •Simplifying complex fractions with variables
        • •Simplifying rational expressions
        • •Simplifying square roots of rational expressions
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        • •Solving rational equations
     › Algebra 2 › Complex numbers

    Solving equations with complex numbers

    Students will learn how to solve equations with complex numbers by equating real and imaginary parts. Here's a lesson.

    Conclude by giving your students these challenges:

    • You Never Get a Six by NRICH
    • 2020 Math Kangaroo Levels 5-6 Problem #24 by STEM4all
    • 2018 AMC 8, Problem 12

    Give your students the following three-part challenge:


    Part 1: Suppose \(D\) occurs \(2/3\) along \(\overline{AB}.\)

    How are the areas of \(\triangle ADC\) and \(\triangle ABC\) related? After you've found the solution, generalize from \(2/3\) to any rational.

    Solution: \(\triangle ADC\) has \(2/3\) the area of \(\triangle ABC.\) That's because the base of \(\triangle ADC\) is \(2/3\) the base of \(\triangle ABC.\)


    Part 2: Suppose \(D\) occurs \(2/3\) along \(\overline{AB},\) \(E\) occurs \(2/3\) along \(\overline{BC},\) and \(F\) occurs \(2/3\) along \(\overline{CA}.\)

    How are the areas of \(\triangle DEF\) and \(\triangle ABC\) related? After you've found the solution, generalize from \(2/3\) to any rational.

    Solution: Suppose \(\triangle ABC\) has area \(1.\) Then \(\triangle ADC\) has area \(2/3,\) as we've already established. Next, notice that \(\triangle CFD\) has \(2/3\) the area of triangle \(ADC,\) by the same reasoning. So \(\triangle CFD\) has area \((2/3) \cdot (2/3) = 4/9.\) If we remove the area of \(\triangle CFD\) from the area of \(\triangle ADC,\) we'll get the area of triangle \(ADF.\) So the area of triangle \(ADF\) must be \(2/3 - 4/9 = 6/9 - 4/9 = 2/9.\) But we could use the exact same reasoning for \(\triangle BEF\) and \(\triangle CFE.\) Since the area of \(\triangle DEF\) is the area of \(\triangle ABC,\) minus the areas of \(\triangle ADF,\) \(\triangle BED,\) and \(\triangle CFE,\) the area of \(\triangle DEF\) must be

    $$1 - \dfrac{3 \cdot 2}{9} = \dfrac{9}{9} - \dfrac{6}{9} = \dfrac{3}{9} = \dfrac{1}{3}$$

    Thus, in summary, the area of \(\triangle DEF\) is \(1/3\) the area of \(\triangle ABC.\) For another elegant solution, go here and read the solution by Miranda and Sadaf from Greenacre Public School in Australia. It's very clever, and thus, it's unlikely your students will find this solution on their own. Even if they don't find it, you should still demonstrate it.


    Part 3: Suppose we continue this pattern. That is, going counterclockwise around each triangle, we make a point \(2/3\) along each segment, and connect each point to form a new triangle. Suppose we do this \(n\) times, and label the innermost triangle \(\triangle DEF.\)

    How are the areas of \(\triangle DEF\) and \(\triangle ABC\) related? After you've found the solution, generalize from \(2/3\) to any rational.

    Solution: The first inner triangle will be \(1/3\) the area of \(\triangle ABC.\) By the same reasoning, the second inner triangle will be \(1/3\) the area of the first inner triangle, that is, \((1/3)^2.\) Thus, the \(n\)-th inner triangle will be \((1/3)^n\) the area of \(\triangle ABC.\)


    Note: This challenge was inspired by the NRICH challenge Triangle in a Triangle.

    Lessons and practice problems