I got the idea from ["The cartoon guide to ALGEBRA"] (https://www.amazon.com/Cartoon-Guide-Algebra/dp/0062202693) to write this article.
One of the interesting things when learning Math is we can convert equations to shapes or graphs. Doing that we also get new aspect of the equations and find new idea.
In this article, I tried to proof quadratic formula that I already forgot, despite I used more than tens of times at school days. Using shapes is very useful to remember how to solve the quadratic equation even though I forget the formula.
Let's see the equation below.
We can use the distributive law like this.
Also the equation is converted to "area of a rectangle" that has a length of (a+b) and a width of (c+d) . To find the area of a rectangle, multiply the length by the width.
If you divided the 1 big rectangle to small 4 rectangles, you can find the area of big rectangle is same as sum of 4 small rectangles area.
Now we prove the distributive law.
Look at the quadratic equation below.
If using factorization, we can easily find the solution. But I'm going to use shapes to solve it. Let's convert the equation. You can find the rectangle that has a length of (x+6) and a width of x.
The area of the rectangle is x(x+6). Some one might ask "how the area of rectangle can be negative?". Good Question!! But it's very difficult question for me(my major is not math). My answer is this. You can imagine anything and everything is possible in math, as a great mathematician find "imaginary number".
Now we need scissors and glue. Cut the rectangle in right half and paste it below the left rectangle. It just cut and paste, so the area of the shape still x(x+6).
Look at the shape with birds eye, you can find beautiful square with each side measuring (X+3). The shape that we made is part of that square. Please do not forget the area of our shape still X(X+6).
We can calculate the area of our shape in another perspective. Area of big square with (X+3) minus area of small square with 3. Now we and make another equation.
Let's assign [수식2] to [수식1]. (수식 means equation in Korean)
X is -2 and -4
When , find the X.
To make rectangle, coefficient of X square should be 1.
And make rectangle.
Cut and paste the right half rectangle.
As we already know, the area of our shapes is big square minus small square.
That means this.
Let's assign [수식2] to [수식1].
It's your turn. Solve for x.
As you already know, to solve the quadratic equation we should make a rectangle to square. Square is beautiful so we can easily get the answer.
Goal : To solve the quadratic equation, make it square.
first : Make equation to "area of rectangle"
second : Make "rectangle to square"
final : Solve for X
