Download Lecture 5: Worksheets and more Exams Geometry in PDF only on Docsity! E-320: Teaching Math with a Historical Perspective Oliver Knill, 2015 Lecture 5: Worksheets We look at the quadratic and the cubic equation and then at puzzles like the 15 puzzle or Rubik type puzzles. The quadratic equation The solution of the quadratic equation x2+bx+c = 0 is one of the major achievements of early algebra. It relies on the method of completion of the square and is due to the Persian mathematician Al Khwarizmi. Artist rendering of Al Khwarizmi for an advertisement. The completion of the square is the idea to add b2/4 on both sides of the equation and move the constant to the right. Like this x2 + bx + b2/4 becomes a square (x + b/2)2. Geometrically, one has added a square to a region to get a square. From (x+ b/2)2 = −c+ b2/4 we can solve x and get the famous formula for the solution of the quadratic equation x = √ b2 4 − c− b 2 . Since one can take both the positive and the negative square root, there are two solutions. x b2 x b2 1) Write down the solution formula for the equation ax2 + bx+ c = 0. 2) If x1, x2 are the two solutions to x 2+bx+c = 0, then the sum of the two solutions is x1+x2 = −b. 3) If x1, x2 are the two solutions of x2 + bx+ c, then the product of the solutions is x1x2 = c. 4) What are the solutions to is x4 − 4x2 + 3 = 0? 5) Find the solutions to x6 − 4x4 + 3x2 = 0. The cubic equation 1) Lets look at the cubic equation x3 − 7x+ 6. Can you figure out the roots? 2) Verify that if a, b, c are solutions to a cubic equation satisfy a + b + c = 0 if and only if it is depressed: x3 + px+ q = 0. Hint: Write (x− a)(x− b)(x− c). Lecture 5: Symmetry groups We look at all the rotational symmetries of a square and realize it as a group. Then, we do the same for all rotational and reflection symmetries of a rectangle. The rotation symmetries of a square Given a square in the plane centered at the origin. We can rotate the square by 90, 180 or 270 degrees and get the same shape. Given two such rotations, we can perform one after the other and get an other rotation. All the rotations leaving the square invariant form a group: one can ”add” these operations and get a new operation. + turn 0 turn 90 turn 180 turn 270 turn 0 turn 0 turn 90 turn 180 turn 270 turn 90 turn 90 turn 180 turn 270 turn 0 turn 180 turn 180 turn 270 turn 0 turn 90 turn 270 turn 270 turn 0 turn 90 turn 180 We can write the multiplication table in a more compact way by writing 1 for the turn 90 degrees, 2 for the turn 180 degrees and 3 for the turn 270 degrees:
