Why is (a + b)² not a² + b²?
Updated
The short answer
(a + b)² means (a + b)(a + b), and multiplying that out gives a² + 2ab + b². Writing a² + b² leaves out the middle term, 2ab, which comes from the two cross products a × b and b × a.
What it looks like
| Wrong | Right |
|---|---|
| (x + 3)² = x² + 9 | (x + 3)² = x² + 6x + 9 |
| (x − 5)² = x² − 25 | (x − 5)² = x² − 10x + 25 |
Numbers make the gap easy to see. (1 + 3)² = 4² = 16, but 1² + 3² = 1 + 9 = 10. The missing 6 is the middle term, 2 × 1 × 3. Paul Dawkins's list of common algebra errors uses this same check.
In alai, an iPad app that reads your handwriting as you work, the tutor, Ally, asks about the first line that doesn't follow from the one before.
Why it happens
The slip looks like distributing. 2(a + b) = 2a + 2b is true, so it feels as if the square should spread over the a and the b the same way. But squaring doesn't multiply each term by something. It multiplies the whole expression by itself:
(x + 3)² = (x + 3)(x + 3) = x² + 3x + 3x + 9 = x² + 6x + 9
Each term in the first parentheses multiplies each term in the second, which makes four products. Two of them are the squares you expect, x² and 9. The other two, 3x and 3x, are the cross products, and together they make the middle term.
A picture helps too. A square with side a + b splits into a square with side a, a square with side b, and two rectangles that measure a by b. Those two rectangles are the 2ab.
Exponents do spread over multiplication, so (ab)² = a²b². They don't spread over addition. Dawkins calls this family of slips additive assumptions, and it includes √(x + y), which isn't √x + √y either.
How to catch it yourself
Whenever a square sits on parentheses, write it out as two sets of parentheses before you multiply. Then point at the little 2 and ask yourself:
What does the 2 on the parentheses ask you to multiply?
Count your products before you combine them. Two sets of parentheses with two terms each should give four products, so if your answer has only two terms, the middle term is missing. For a quick check, put x = 1 into the original and into your answer and see whether they match.
Try these
Multiply out each one.
- (x + 4)²
- (y − 6)²
- (2x + 1)²
Answers
- x² + 8x + 16
- y² − 12y + 36
- 4x² + 4x + 1
How alai helps
In Homework help, when you pause after writing a step, Ally reads it and checks it against the line before. Ally asks about the first line that doesn't follow, with a hint that stops one step short, so the fix is still yours to make. Ally also remembers slips from past sessions, so a later session can start from there.