In this page we practice using the geometric derivation of:

\((a+b)^2=a^2+2ab+b^2\)

We also practice expanding algebraic expressions and verifying the identity using numbers and areas.

Example 1 – Expand the Expression

Expand:

\((x+2)^2\)

Using:

\((a+b)^2=a^2+2ab+b^2\)

we get:

\(x^2+2(x)(2)+2^2\)

\(x^2+4x+4\)

Example 2 – Numerical Example

Evaluate:

\((5+3)^2\)

Using the identity:

\(5^2+2(5)(3)+3^2\)

\(25+30+9\)

\(64\)

So:

\((5+3)^2=64\)

Example 3 – Verify Both Sides

Check whether:

\((6+2)^2=62+2(6)(2)+2^2\)

Left side:

\(8^2=64\)

Right side:

\(36+24+4=64\)

Both sides are equal.

Example 4 – Geometric Idea

A square has side length:

\(a+b\)

where:

\(a=5,b=2\)

The square is divided into:

• one square with area \(a^2\)
• two rectangles with area \(ab\)
• one square with area \(b^2\)

Using the identity:

\(5^2+2(5)(2)+2^2\)

\(25+20+4\)

\(49\)

So:

\((5+2)^2=49\)

Example 5 – Simplify

Expand:

\((y+1)^2\)

Using the identity:

\(y^2+2(y)(1)+1^2\)

\(y^2+2y+1\)

Practice

• More Examples

• Take the Quiz

Continue Learning

1. \((a – b)^2\) – Geometric Derivation
2. \((a + b)^2\) – Geometric Derivation
3. \(a^2 – b^2\) – Geometric Derivation
4. \(a^2 – b^2\) – Algebraic Proof

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