In this page we practice proving and simplifying expressions using:
\(a^2 – b^2 = (a+b)(a-b)\)
We also practice expanding expressions using the distributive law.
Example 1 – Expand the Expression
Expand:
\((x+5)(x−5)\)
Using distributive law:
\(x(x−5)+5(x−5)\)
\(x^2−5x+5x−25\)
The middle terms cancel:
\(x^2−25\)
Example 2 – Another Example
Expand:
\((y+2)(y−2)\)
Distribute:
\(y^2−2y+2y−4\)
Simplify:
\(y^2−4\)
Example 3 – Verify the Identity
Check whether:
\((8+1)(8−1)=8^2−1^2\)
Left side:
\(9×7=63\)
Right side:
\(64−1=63\)
Both sides are equal.
Example 4 – Simplify
Simplify:
\((a+10)(a−10)\)
Using the identity:
\(a^2−10^2\)
\(a^2−100\)
Example 5 – Numerical Example
Evaluate:
\(15^2−5^2\)
Using the identity:
\((15+5)(15−5)\)
\(20 \times 10=200\)
Check directly:\(225−25=200\)
Both methods give the same answer.
Practice
- More Examples
Continue Learning
- \((a – b)^2\) – Geometric Derivation
- \((a + b)^2\) – Geometric Derivation
- \(a^2 – b^2\) – Geometric Derivation
- \(a^2 – b^2\) – Algebraic Proof