Explanation
The algebra identity
\((a-b)^2 = a^2 -2ab + b^2\)
is one of the most common identities in algebra.
In this lesson, we explain this identity geometrically using areas.
We start with a large square of side length:
\(a\)
So the area of the whole square is:
\(a^2\)
Now we cut strips of width:
\(b\)
from the top and the side of the square.
The remaining yellow square has side length:
\(a−b\)
So its area is:
\((a−b)^2\)
To find this area, we subtract the other pieces from the large square.
The removed pieces are:
- one small square with area \(b^2\)
- two rectangles, each with area \(b(a−b)\)
So:
\((a−b)^2=a^2−[b^2+b(a−b)+b(a−b)]\)
Simplify:
\((a−b)^2=a^2−[b^2+2b(a−b)]\)
Expand:
\((a−b)^2=a^2−[b^2+2ab−2b^2]\)
\((a−b)^2=a^2−2ab+b^2\)
So we obtain the identity:
\((a−b)^2=a^2−2ab+b^2\)
Formula / Rule
Area of a Rectangle
Area \(=\) length \(\times\) width
Area of a Square
Area \(=\) side \(\times\) side
Algebra Identity
\((a-b)^2 = a^2 -2ab + b^2\)
Example
Suppose:
\(a=7, b=2\)
Using the identity:
\((a−b)^2=a^2−2ab+b^2\)
Substitute the values:
\((7−2)^2=7^2−2(7)(2)+2^2\)
Simplify:
\(5^2=49−28+4\)
\(25=25\)
Both sides are equal, so the identity is correct.
Video Explanation
Practice
Continue Learning
- \((a – b)^2\) – Geometric Derivation
- \((a + b)^2\) – Geometric Derivation
- \(a^2 – b^2\) – Geometric Derivation
- \(a^2 – b^2\) – Algebraic Proof