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Square of a SUM

MATH LESSON

SQUARE OF A SUM

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

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

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

1st term: a² — square the first term
2nd term: 2ab — multiply both terms, then double it
3rd term: b² — square the second term
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 Mistake!

Many students think (a + b)² = a² + b². This is WRONG! You're forgetting the middle term 2ab.

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

Shortcut: Middle term = 2 × first × second. Last term = second². First term = first²

LEARNING CHECK

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MATH QUIZ

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