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:

\((x−3)^2\)

Using:

\((a−b)^2=a^2−2ab+b^2\)

we get:

\(x^2−2(x)(3)+3^2\)

\(x^2−6x+9\)

Example 2 – Numerical Example

Evaluate:

\((10−4)^2\)

Using the identity:

\(10^2−2(10)(4)+4^2\)

\(100−80+16\)

\(36\)

So:

\((10−4)^2=36\)

Example 3 – Verify Both Sides

Check whether:

\((8−5)^2=8^2−2(8)(5)+5^2\)

Left side:

\(3^2=9\)

Right side:

\(64−80+25=9\)

Both sides are equal.

Example 4 – Geometric Idea

A large square has side:

\(a=6\)

A strip of width:

\(b=2\)

is removed from the top and side.

The remaining square has side:

\(a−b=4\)

So its area is:

\(4^2=16\)

Using the identity:

\(6^2−2(6)(2)+2^2\)

\(36−24+4\)

\(16\)

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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