Math Doubts

Proof of a2 – b2 formula in Geometric Method

Formula

$a^2-b^2 \,=\, (a+b)(a-b)$

The $a^2-b^2$ identity represents the difference of two square quantities and it can be written in factoring form as the product of binomials $a+b$ and $a-b$. The factoring form of $a^2-b^2$ formula can be derived in mathematics geometrically on the basis of areas of geometric shapes.

Subtracting areas of squares

a squared subtracted b squared
  1. Take a square, whose length of each side is $a$ units. Therefore, the area of the square is $a^2$.
  2. Draw a small square with the side of $b$ units at any corner of the square. So, the area of small square is $b^2$.
  3. Now, subtract the square, whose area is $b^2$ from the square, whose area is $a^2$. It forms a new geometric shape and its area is equal to $a^2-b^2$.

Dividing Area of New shape

a squared minus b squared equivalent form
  1. Divide the new subtracted geometric shape as two different rectangles but the length of one of the two rectangles should be equal to $b$ units.
  2. Look at the upper rectangle. Geometrically, the length of this rectangle is $a$ units and its width is $a-b$ units.
  3. Similarly, look at the lower rectangle. The length of this rectangle is $a-b$ units and its width is equal to $b$ units.

Difference of Squares in factoring form

The width of the upper rectangle is $a-b$ and the length of the lower rectangle is also $a-b$ geometrically. If the lower rectangle is rotated by $90^\circ$, then the widths of both rectangles become same and it is useful to join them together as a rectangle.

factoring a squared minus b squared
  1. Separate both rectangles.
  2. Rotate the lower rectangle by $90^\circ$ and then join both rectangles. It formed another rectangle.
  3. The length and width of the new rectangle are $a+b$ and $a-b$ respectively. Therefore, the area of this rectangle is $(a+b)(a-b)$.

In first step, it is derived that the area of subtracted shape is $a^2-b^2$ and the same shape is now transformed as a rectangle, whose area is ${(a+b)}{(a-b)}$.

Therefore, the areas of both shapes should be equal geometrically.

$\,\,\, \therefore \,\,\,\,\,\, a^2-b^2 \,=\, (a+b)(a-b)$

Geometrically, it is proved that the $a^2$ subtracted $b^2$ is equal to the product of the binomials $a+b$ and $a-b$.

Math Doubts

A best free mathematics education website for students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved