HighSchoolQ

➕ Algebra 1 · Factoring

Difference of Squares: notes and common mistakes

Study notes

Key ideas

  • A difference of squares is a binomial with one perfect square subtracted from another. It factors as a2 − b2 = (a + b)(a − b).
  • Factoring is the reverse of multiplying. Check by expanding: the middle terms −ab and +ab cancel.
  • A term is a perfect square if its coefficient is a perfect square and every variable exponent is even. Root the number and halve the exponents.
  • Take out any GCF first. Then keep factoring until the expression is completely factored: x4 − 16 = (x2 + 4)(x + 2)(x − 2).
  • After any GCF is taken out, a two-term sum of squares of degree 2, such as x2 + 9, is prime: it cannot be factored using integers, or even with real-number coefficients.
  • The pattern also gives mental math: (n + d)(n − d) = n2 − d2.

Steps

  1. F: take out any GCF first.
  2. A: confirm there are two terms, with one perfect square subtracted from another.
  3. C: choose a and b (root the numbers, halve the exponents).
  4. T: write two factors, (a + b)(a − b).
  5. S: spot-check by expanding, then scan each factor to see whether it factors again.

Formulas

  • Difference of squares: a2 − b2 = (a + b)(a − b)
  • Expanding the check: (a + b)(a − b) = a2 − ab + ab − b2 = a2 − b2
  • Mental math: (n + d)(n − d) = n2 − d2
  • A square of a power: (xk)2 = x2k, so halve the exponent to find the root

Watch out

  • A plus sign is not a difference. x2 + 9 cannot be factored using integers.
  • x2 − 9 = (x + 3)(x − 3), not (x − 3)2. The two factors have opposite signs.
  • Take the square root of the coefficient too: 25x2 − 64 = (5x + 8)(5x − 8).
  • Look for a GCF first. 18x2 − 50 has no perfect-square terms until the 2 comes out.
  • Stop only when no factor can be factored further. (x2 + 4)(x2 − 4) is not complete.
  • The pattern needs both terms to be perfect squares. x2 − 10 does not factor using integers.

Why it works

  • Why does the pattern work? Multiplying (a + b)(a − b) gives the middle terms −ab and +ab, which cancel, leaving a2 − b2.
  • Why does a two-term sum of squares like x2 + 9 not factor? Cancelling the middle term needs opposite signs, which makes the last term negative. And any factor (x + p) would make the product 0 at x = −p, but x2 + 9 is never 0 for real x.
  • Why does the mental-math trick work? A pair like 47 × 53 is a sum and a difference around 50, so it equals 502 − 32.

Common mistakes

✗ Mistakex2 − 25 = (x − 5)2 (using the same sign twice).
✓ Instead(x − 5)2 = x2 − 10x + 25 has a middle term. A difference of squares has opposite signs in its factors: x2 − 25 = (x + 5)(x − 5).
✗ Mistakex2 + 16 = (x + 4)(x − 4) (a sum of squares factors like a difference).
✓ Instead(x + 4)(x − 4) = x2 − 16. A two-term sum of squares such as x2 + 16 is prime: it cannot be factored using integers.
✗ Mistake4x2 − 81 = (4x + 9)(4x − 9) (the coefficient is not rooted).
✓ InsteadTake the square root of the coefficient too: 4x2 = (2x)2, so 4x2 − 81 = (2x + 9)(2x − 9). The wrong answer expands to 16x2 − 81.
✗ Mistakex4 − 16 = (x2 + 4)(x2 − 4) is completely factored.
✓ Insteadx2 − 4 is a difference of squares and factors again. The complete answer is (x2 + 4)(x + 2)(x − 2).
✗ Mistake18x2 − 50 cannot be factored, because 18 and 50 are not perfect squares.
✓ InsteadTake out the GCF first: 18x2 − 50 = 2(9x2 − 25) = 2(3x + 5)(3x − 5).

Teach it back

Explain to a friend how to factor 45x2 − 20 completely, why you look for a GCF first, and why x2 + 4 cannot be factored.

Explaining it out loud shows what you really know. Did you…

  • I took out the GCF 5 first, getting 5(9x2 − 4), because 45 and 20 are not perfect squares.
  • I found a = 3x and b = 2, wrote 5(3x + 2)(3x − 2), and checked by expanding: (3x + 2)(3x − 2) = 9x2 − 4 and 5(9x2 − 4) = 45x2 − 20.
  • I explained that a sum of squares like x2 + 4 is prime: opposite signs would make a minus sign, and equal signs would make a middle term.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

Downloads a calendar file (.ics). Works with Google, Apple and Outlook calendars. Nothing is sent to us.