Multiplication Formulas
“The secret of (a+b)² is the 2ab wedged in the middle.”
The formula
(a + b)² = a² + 2ab + b²How to read it: the square of a sum is each square plus 2ab
- a, b
- — the two terms being added
- 2ab
- — the middle term that's easy to forget
- (a + b)²
- — the area of a square with side a+b
The hook
No need to multiply (a+b)² out by hand every time — one area picture burns the pattern into your eyes.
In plain words
(a+b)² is not a²+b²; a 2ab squeezes into the middle, making it a²+2ab+b².
The intuition
Draw a square with side a+b. It splits into four pieces — an a² square, a b² square, and 'two' a×b rectangles. Those two rectangles are exactly the 2ab.
How it's built
Three core formulas — (a+b)²=a²+2ab+b², (a−b)²=a²−2ab+b², and (a+b)(a−b)=a²−b². The first two have the 'middle 2ab'; the last one loses the middle and leaves a difference of squares.
Example
(x+3)² = x² + 2·x·3 + 3² = x²+6x+9. That middle 6x is the 2ab slot.
Common mistake
(a+b)² ≠ a²+b². Forgetting the 2ab is algebra's single most common slip. There is always a middle term.
Where it's used
It collapses messy multiplication into one line and speeds up mental math — and it's the key you read backward to do the next step, factoring.
Where it came from
These relationships appear in Book II of Euclid's 'Elements' as geometric theorems about 'splitting areas', complete with diagrams.
Prerequisites
Quick check
Expand (x + 2)².
- x² + 4x + 4✓
- x² + 4
- x² + 2x + 4
- x² + 4x + 2
Practice
Use a multiplication formula to compute 102².
Answer: 10404
- 102² = (100 + 2)²
- = 100² + 2·100·2 + 2² = 10000 + 400 + 4
- = 10404
Takeaway: The (a+b)² formula squares big numbers in your head.
Use a multiplication formula to compute 21 × 19.
Answer: 399
- 21 × 19 = (20 + 1)(20 − 1)
- = 20² − 1² = 400 − 1
- = 399
Takeaway: A sum times a difference is a²−b² — the middle vanishes.
Expand (x + 5)².
Answer: undefined
- (x+5)² = x² + 2·x·5 + 5²
- = x² + 10x + 25
Takeaway: Don't forget the middle term 10x (=2ab).