THE QUADRATIC EQUATION

We look at the quadratic and the cubic equation and then at puzzles like the 15 puzzle or Rubik type puzzles. The solution of the quadratic equation x 2 + 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 b 2 /4 on both sides of the equation and move the constant to the right. Like this x 2 + bx + b 2 /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 + b 2 /4 we can solve x and get the famous formula for the solution of the quadratic equation x = b 2 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 ax 2 + bx + c = 0. 2) If x 1 , x 2 are the two solutions to x 2 +bx+c = 0, then the sum of the two solutions is x 1 +x 2 = −b. 3) If x 1 , x 2 are the two solutions of x 2 + bx + c, then the product of the solutions is x 1 x 2 = c.