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A Square of Things Quadratic Equations PowerPoint PPT Presentation


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A Square of Things Quadratic Equations. By: Ellen Kramer. Year 825: Muhammad Ibn Musa Al-Khwarizmi wrote Arabic book titled “algebra”. Discusses the quadratic equation with a specific problem:

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A Square of Things Quadratic Equations

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A square of things quadratic equations l.jpg

A Square of ThingsQuadratic Equations

By: Ellen Kramer


Algebra from the beginning l.jpg

Year 825: Muhammad Ibn Musa Al-Khwarizmi wrote Arabic book titled “algebra”

Discusses the quadratic equation with a specific problem:

“one square, and ten roots of the same, are equal to thirty-nine…what must be the square which, when increased by ten of its own roots, amounts to thirty-nine?”

Algebra from the Beginning


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Solutions in 825

  • No algebraic symbolism, thus all problems are like recipe cards

    • Solution: “you halve the number of the roots, which in the present instance yields five. This you multiple by itself; the product is twenty-five. Add this to thirty-nine; the sum is sixty-four. Now take the root of this, which is eight, and subtract from it half the number of the roots, which is five; the remainder is three. This is the root of the square which you sought for; the square itself is nine.

Quadratic formula:

X= b 2 b

+ c -

2 2


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Solutions Used Today

  • Early 17th Century mathematicians came up with algebraic symbols

    • Letters from the end = unknown numbers

      • Example: x, y, z

    • Letters from the beginning = known numbers

      • Example: a, b, c

    • Thomas Harriot and Rene Descartes rearranged equations so that they always equal 0.

      • Thus: ax2 + bx = c & ax2 + c = bx

        Became ax2 + bx + c = 0


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Solutions Today Cont.

Question: “one square, and ten roots of the same, are equal to thirty-nine…what must be the square which, when increased by ten of its own roots, amounts to thirty-nine?

  • Translate:

    • Unknown: x “root of the square x2 “

    • “ten roots of the square”  10x

      • Equation: x2 + 10x = 39

  • Solution: “you halve the number of the roots, which in the present instance yields five. This you multiple by itself; the product is twenty-five. Add this to thirty-nine; the sum is sixty-four. Now take the root of this, which is eight, and subtract from it half the number of the roots, which is five; the remainder is three.”

    • Compute:

    • 52 + 39 - 5 =

      • 25 + 39 - 5 =

      • 64 - 5 =

      • 8 - 5 = 3

Quadratic formula:

X= -b + b2 + 4c

2


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Explanation of Method Using a Geometric Argument

x

5

x

10

x

x2

5x

x

x2

10x

5

5x

x

5

x

x2

5x

5

5x

25


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Questions?

Quadratic formula:

X= -b + b2 + 4ac

2

Thanks!


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