Square of Binomial
(a+b)²=a²+2ab+b². (a−b)²=a²−2ab+b². (a+b)³=a³+3a²b+3ab²+b³. Perfect square trinomial: b²=4ac. Area model: square splits into a², 2ab, b².
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Middle term: 2ab for (a+b)², −2ab for (a−b)². Mental math: 23²=(20+3)²=400+120+9=529. Perfect square trinomial factors as (√a·x±√c)².
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Why: Binomial squares appear in completing the square, factoring, and mental math. 25²=(20+5)²=400+200+25=625. Geometric: area of (a+b)×(a+b) square.
How: (a+b)²: square first, twice product, square last. (a−b)²: same but middle −2ab. (a+b)³: Pascal triangle 1,3,3,1. Verify: check if ax²+bx+c has b²=4ac.
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Term Values
Term Contribution
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🧮 Fascinating Math Facts
(x+3)²=x²+6x+9. Middle 6=2·x·3, last 9=3².
— Example
23²=(20+3)²=400+120+9=529. Mental math.
— Mental Math
📐 Geometric Proof (Square Area)
Draw a square of side (a+b). It splits into: one a×a square, two a×b rectangles, one b×b square. So (a+b)² = a² + 2ab + b². For (a-b)², the area is a² - 2ab + b² (the two rectangles are "removed").
🔗 Connection to Completing the Square
To complete x² + bx, add (b/2)² to get (x + b/2)². This uses the identity (a+b)² = a² + 2ab + b² with a=x, b=b/2.
⚠️ Common Mistake: (a+b)² ≠ a² + b²
Many students forget the middle term 2ab. (a+b)² = a² + 2ab + b². Example: (3+4)² = 49, but 3²+4² = 25. The 2ab = 24 makes the difference!
🧮 Mental Math Tricks
To square numbers near 100: 101² = (100+1)² = 10000 + 200 + 1 = 10201. 97² = (100-3)² = 10000 - 600 + 9 = 9409.
📋 Cube of Binomial
(a+b)³ = a³ + 3a²b + 3ab² + b³. The coefficients 1, 3, 3, 1 come from Pascal's triangle row 3.
📝 Worked Examples
📌 Summary
(a+b)² = a² + 2ab + b² and (a-b)² = a² - 2ab + b². Never forget the middle term! Use for completing the square, factoring perfect square trinomials, and mental math (e.g. 97² = (100-3)²).
❓ FAQ
What is (a+b)²?
(a+b)² = a² + 2ab + b². Square the first, double the product, square the second.
What is (a-b)²?
(a-b)² = a² - 2ab + b². Same as sum but the middle term is negative.
Why is (a+b)² not a² + b²?
Expanding (a+b)(a+b) gives a² + ab + ba + b² = a² + 2ab + b². The 2ab comes from both cross terms.
How do I factor x²+10x+25?
It equals (x+5)² because 25=5² and 10=2·x·5. Perfect square trinomial.
What is (a+b)³?
(a+b)³ = a³ + 3a²b + 3ab² + b³. Coefficients 1,3,3,1 from Pascal's triangle.
How does this connect to completing the square?
x² + bx + (b/2)² = (x + b/2)². We add (b/2)² to create a perfect square.
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