NISHIO Hirokazu

4 * Sum(x^3) = (n * (n + 1))^2

2009-05-21 published on my blog: Σ(x^3) = (Σx)^2
2009-06-03 published painted version here.

I made another version to make me clear. Now you see it is 4 * (1 * (1 * 1) + 2 * (2 * 2) + ...), (2 * (1 + 2 + ...))^2 and ((5 * 5) + 5)^2 simaltaneously. Thus 4 * Sum(x^3) = (2 * Sum(x))^2 = (n * (n + 1))^2.

solutions of quadratic equation

2009-04-23 published in my blog: 二次関数の解的な何か解説編

rectangles' width

x

rectangles' height

an constant b

area of rectangles

bx

area of white square in rectangles

x^2

area of gray rectangle in rectangles

See it is constant. It is c

Thus there is an equation bx = x^2 + c. It means x^2 - bx + c = 0.

Now let's see the right side.

height of large square

A half of the height b. b / 2.

area of large square

(b / 2)^2. b^2 / 4.

gray area

constant c

area of small square

It rests after you subtract gray area from large square. So it is b^2 / 4 - c.

height of large square

sqrt(b^2 / 4 - c) = sqrt(b^2 - 4 * c) / 2

height of lefthand square x

b / 2 +- sqrt(b^2 - 4 * c) / 2

You know x = (b +- sqrt(b^2 - 4 * c)) / 2 is a solution of quadratic equation x^2 - bx + c = 0.

a^2 - b^2 = (a - b) * (a + b)

2009-04-22 published in my blog: a^2 - b^2 = (a - b) * (a + b)