Descarte Quadratic

Reneé Descarte, the French mathematician, proved that a quadratic equation in the form x^2+a*x=b^2 could be solved geometrically.

Construction of the Decarte Quadratic

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The figure is constructed as follows:

  1. Draw a line segment AB of length b.
  2. Draw a line segment AC of length a/2 perpendicular to AB.
  3. Draw a line segment CB.
  4. With C as a center, construct a circle of radius a/2.
  5. Draw point E at the intersection of the circle and the segment AB.
  6. You can now measure the length of segment EB. This is the value of x.

Proof

  1. Since CA and AE are both radii of the circle, they are both of length a/2.
  2. By the pythagorean theorem, (a/2)^2 + b^2 = (x + a/2)^2
  3. This equation can be simplified as follows:
    1. a^2/4 + b^2 = x^2 + ax + a^2/4 (do operations inside of parenthesis and expand the quadratic.
    2. a^2/4 - a^2/4 + b^2 = x^2 + ax + a^2/4 - a^2/4 (subtract a^2/4 from both sides).
    3. b^2 = x^2 + ax
  4. QED