Math Doubts

Proof of a2 – b2 formula in Geometric Method


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

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 Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.

Learn more

Math Doubts

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved