In this page we practice using the geometric derivation of:
\(a^2−b^2=(a+b)(a−b)\)
We also practice simplifying expressions and verifying the identity using numbers and areas.
Example 1 – Numerical Example
Evaluate:
\(9^2−4^2\)
Using the identity:
\((9+4)(9−4)\)
\(13 \times 5=65\)
Check directly:
\(81−16=65\)
Both methods give the same answer.
Example 2 – Simplify
Simplify:
\(x^2−25\)
Notice:
\(25=5^2\)
So:
\(x^2−25=(x+5)(x−5)\)
Example 3 – Another Example
Simplify:
\(y^2−16\)
Since:
\(16=4^2\)
we get:
\((y+4)(y−4)\)
Example 4 – Geometric Idea
A large square has side length:
\(a=8\)
A smaller square inside has side length:
\(b=2\)
The remaining area is:
\(8^2−2^2=64−4=60\)
Using the identity:
\((8+2)(8−2)=10×6=60\)
Both methods give the same area.
Example 5 – Verify the Identity
Check whether:
\(6^2−1^2=(6+1)(6−1)\)
Left side:
\(36−1=35\)
Right side:
\(7 \times 5=35\)
Both sides are equal.
Practice
- More Examples (Current page)
Continue Learning
- \((a – b)^2\) – Geometric Derivation
- \((a + b)^2\) – Geometric Derivation
- \(a^2 – b^2\) – Geometric Derivation
- \(a^2 – b^2\) – Algebraic Proof