Explanation
The algebra identity
\(a^2 – b^2 = (a+b)(a-b)\)
is called the difference of two squares identity.
In this lesson, we prove this identity algebraically using the distributive law.
We start from the right-hand side:
\((a+b)(a−b)\)
Now multiply each term:
\(a(a−b)+b(a−b)\)
Distribute again:
\(a^2−ab+ab−b^2\)
Notice that:
\(−ab+ab=0\)
So the middle terms cancel:
\(a^2−b^2\)
Therefore:
\(a^2−b^2=(a+b)(a−b)\)
This proves the identity algebraically.
Formula / Rule
Distributive Law
\(x(y+z)=xy+xz\)
Difference of Two Squares Identity
\(a^2 – b^2 = (a+b)(a-b)\)
Example
Prove algebraically that:
\(x^2−9=(x+3)(x−3)\)
Start from the right-hand side:
\((x+3)(x−3)\)
Distribute:
\(x(x−3)+3(x−3)\)
\(x^2−3x+3x−9\)
The middle terms cancel:
\(−3x+3x=0\)
So:
\(x^2−9\)
Therefore:
\(x^2−9=(x+3)(x−3)\)
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