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Square of a Difference

MATH LESSON

SQUARE OF A DIFFERENCE

Master the formula: (a − b)² = a² − 2ab + b²

(a − b)² = a² − 2ab + b²
Understanding the Formula

The square of a difference means you multiply (a − b) by itself. The result is a perfect square trinomial with 3 terms:

1st term: a² — square the first term
2nd term: −2ab — multiply both, double it, make it negative
3rd term: +b² — square the second term (always positive)
Where Does −2ab Come From?

When you use FOIL on (a − b)(a − b), the Outer and Inner products are both −ab. Adding them gives −2ab!

(a − b)(a − b)
F: a · a = a²
O: a · (−b) = −ab
I: (−b) · a = −ab
L: (−b) · (−b) = +b²
Combine O + I: −ab + (−ab) = −2ab
Final: a² − 2ab + b²
Common Mistakes!

Mistake 1: Forgetting the middle term entirely.

(x − 3)² = x² − 9 × WRONG
(x − 3)² = x² − 6x + 9 ✓ CORRECT

Mistake 2: Making the last term negative.

(x − 3)² = x² − 6x − 9 × WRONG
(x − 3)² = x² − 6x + 9 ✓ CORRECT

Remember: Negative × Negative = Positive! The last term is always positive.

Sum vs. Difference — Spot the Difference!

The ONLY difference is the sign of the middle term. First and last terms are always the same!

SQUARE OF A SUM
(a + b)² = a² + 2ab + b²
Middle term is POSITIVE
SQUARE OF A DIFFERENCE
(a − b)² = a² − 2ab + b²
Middle term is NEGATIVE
Worked Examples
Example 1: (x − 5)²
a = x → x² = x²
−2ab = −2(x)(5) = −10x
b = 5 → 5² = 25
→ x² − 10x + 25
Example 2: (2x − 3)²
a = 2x → (2x)² = 4x²
−2ab = −2(2x)(3) = −12x
b = 3 → 3² = 9
→ 4x² − 12x + 9
Example 3: (3x − 4)²
a = 3x → (3x)² = 9x²
−2ab = −2(3x)(4) = −24x
b = 4 → 4² = 16
→ 9x² − 24x + 16
Example 4: (x − 10)²
a = x → x²
−2ab = −2(x)(10) = −20x
b = 10 → 10² = 100
→ x² − 20x + 100

Key Trick: Same first & last terms as (a+b)², but middle term is NEGATIVE. Last term is ALWAYS positive!

LEARNING CHECK

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MATH QUIZ

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