Square of a Binomial: Formula, Geometric Proof, and Common Mistakes
Squaring a binomial is one of the most frequent operations in algebra. Rather than carrying out tedious foil multiplication each time, applying the binomial square identity produces instant results.
The Special Product Formulas
(ax + b)^2 = (ax)^2 + 2(ax)(b) + b^2 = a^2 x^2 + 2ab\,x + b^2
(ax - b)^2 = (ax)^2 - 2(ax)(b) + b^2 = a^2 x^2 - 2ab\,x + b^2
The Classic "Freshman's Dream" Error
One of the most persistent errors in mathematics is distributing the exponent over addition:
(a + b)^2 \ne a^2 + b^2
Omitting the middle term $2ab$ ignores the two cross-product rectangles!
Geometric Area Proof
Imagine a large square of side length $(a + b)$. Its total area $(a + b)^2$ is partitioned into four sub-regions:
- 1One large square of area $a \times a = a^2$.
- 2Two identical rectangles of area $a \times b = ab$.
- 3One small square of area $b \times b = b^2$.
Summing these four pieces gives $a^2 + 2ab + b^2$.
Binomial Square Reference Table
| Binomial Expression | First Squared | Twice Product | Last Squared | Expanded Trinomial |
|---|---|---|---|---|
| $(x + 5)^2$ | $x^2$ | $2(x)(5) = 10x$ | $5^2 = 25$ | $x^2 + 10x + 25$ |
| $(2x - 3)^2$ | $4x^2$ | $2(2x)(-3) = -12x$ | $(-3)^2 = 9$ | $4x^2 - 12x + 9$ |
| $(3x + 4)^2$ | $9x^2$ | $2(3x)(4) = 24x$ | $4^2 = 16$ | $9x^2 + 24x + 16$ |
| $(5x - 1)^2$ | $25x^2$ | $2(5x)(-1) = -10x$ | $(-1)^2 = 1$ | $25x^2 - 10x + 1$ |