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Why (a + b)2 isn't a2 + b2 — and other distribution mistakes

The most common algebra mistake there is, why it's wrong, and the handful of related slips that come from the same idea.

Short answer

(a + b)2 means (a + b)(a + b), and multiplying that out gives four products, not two: (a + b)2 = a2 + 2ab + b2. Squaring doesn't distribute over addition. The same goes for square roots and other powers.

Check it with numbers

Any time you're unsure whether an algebra rule is real, try small numbers. Take a = 3 and b = 4:

  1. (3 + 4)2 = 72 = 49Add first, then square.
  2. 32 + 42 = 9 + 16 = 25Not 49. The “rule” fails.
  3. 32 + 2(3)(4) + 42 = 49The real expansion: 9 + 24 + 16 = 49. The missing 24 is the 2ab.

One counterexample is enough to show a rule is false. That trick works for every mistake on this page.

See it as area

A square with side a + b. Its area is (a + b)2. The two squares make a2 + b2, but the two rectangles, ab each, are part of it too.

Worked examples

Expand (x + 5)2:

  1. (x + 5)2Start.
  2. (x + 5)(x + 5)Write the square as a product.
  3. x2 + 5x + 5x + 25Multiply each term in the first by each term in the second.
  4. x2 + 10x + 25Combine like terms.

Expand (2x − 3)2. Watch the signs and the coefficient on x:

  1. (2x − 3)(2x − 3)Write the square as a product.
  2. 4x2 − 6x − 6x + 9(2x)2 = 4x2, and (−3)(−3) = +9.
  3. 4x2 − 12x + 9

The three patterns worth memorizing:

Where it happens in a real problem

Solve (x + 3)2 = x2 + 15.

  1. (x + 3)2 = x2 + 15Start.
  2. x2 + 9 = x2 + 15This is the mistake. The 6x middle term is missing.
  3. 9 = 15Leads to “no solution,” which is wrong.

Done correctly:

  1. x2 + 6x + 9 = x2 + 15Expand with the middle term.
  2. 6x + 9 = 15Subtract x2 from both sides.
  3. x = 1Subtract 9, divide by 6.

Check: (1 + 3)2 = 16 and 12 + 15 = 16. The final answer “no solution” gave no hint that the problem was one missing term on the second line.

Other distribution mistakes from the same idea

Distributing a minus sign to only the first term

A minus sign in front of parentheses multiplies every term inside by −1.

  1. 3x − (x − 4) = 10Start.
  2. 3x − x − 4 = 10This is the mistake. −(x − 4) = −x + 4. The −4 should have become +4.
  3. x = 7Checking: 3(7) − (7 − 4) = 18, not 10.
  1. 3x − x + 4 = 10Correct: both signs change.
  2. 2x = 6Combine and subtract 4.
  3. x = 3Check: 9 − (−1) = 10.

Multiplying only the first term

2(x + 3) is 2x + 6, not 2x + 3. The 2 multiplies everything inside the parentheses.

Square roots don't split over addition either

√(a2 + b2) is not a + b. With 3 and 4: √(9 + 16) = √25 = 5, but 3 + 4 = 7. You can split a root over multiplication, √(ab) = √a · √b for non-negative a and b, but not over addition.

What does distribute

Powers do distribute over multiplication: (ab)2 = a2b2 and (2x)2 = 4x2. The mistake is carrying that rule over to addition, where it doesn't hold.

Practice

  1. Expand (x + 4)2.

    Show answer

    x2 + 8x + 16

  2. Expand (3x − 1)2.

    Show answer

    9x2 − 6x + 1. The middle term is 2 · (3x) · (−1) = −6x.

  3. Expand (x + 2)(x − 2).

    Show answer

    x2 − 4. The middle terms +2x and −2x cancel.

  4. Simplify 5 − 2(x − 3).

    Show answer

    5 − 2x + 6 = 11 − 2x. The −2 multiplies the −3 too, giving +6.

  5. True or false: (x + y)3 = x3 + y3.

    Show answer

    False. Try x = 1, y = 1: (1 + 1)3 = 8, but 13 + 13 = 2.

  6. A student writes (x − 1)2 = x2 + 1. Find a number that shows it's wrong.

    Show answer

    Try x = 4: (4 − 1)2 = 9, but 42 + 1 = 17. The correct expansion is x2 − 2x + 1.

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