A messy-looking product becomes clear once you picture it as area.
Split a square of side a+b into four pieces and you get a², ab, ab, and b², so (a+b)² = a² + 2ab + b² appears on its own.
Flip a sign for (a−b)², while sum times difference (a+b)(a−b) loses its middle and leaves just a² − b².
Here you slide a and b to see how the area splits and why each identity holds, then use them on sums like 101² or 51×49.
(a+b)² — Area Model
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Visualize (a+b)² by dividing a square into four regions
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📐 Idea of the Area Model
①A square with side a+b has area (a+b)²
②Cut it into four pieces: a² + ab + ab + b²
③Therefore (a+b)² = a² + 2ab + b²
Perfect-Square Identities
Sum Squared
(a+b)² = a² + 2ab + b²
Square of a sum = each squared + twice product
Difference Squared
(a−b)² = a² − 2ab + b²
Same idea, only the middle sign flips
💡 Sign Tip
①(a+b)²: middle = +2ab
②(a−b)²: middle = −2ab
③End terms a², b² are always positive (they are squares!)
(a+b)(a−b) — Difference of Squares
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Subtracting b² from a² yields (a+b)(a−b)
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Sum × Difference
(a+b)(a−b) = a² − b²
Sum times difference = difference of squares
Applying the Formulas
Numerical Trick
101² = (100+1)² = 10000+200+1 = 10201
Multiplication formulas turn ugly numbers easy
🧮 Examples
①99² = (100−1)² = 10000−200+1 = 9801
②51×49 = (50+1)(50−1) = 2500−1 = 2499
③(√3+1)² = 3+2√3+1 = 4+2√3
Work It Out
Example 1
Expand (x + 3)².
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Use the perfect-square formula (a + b)² = a² + 2ab + b².
(x + 3)² = x² + 2·x·3 + 3²
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Simplify.
= x² + 6x + 9
▸ x² + 6x + 9
Do not drop the middle term 2ab of a perfect square.
Example 2
Expand (x + 2)(x − 5).
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Use (x + a)(x + b) = x² + (a + b)x + ab with a = 2, b = −5.
x² + (2 + (−5))x + 2·(−5)
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Simplify.
= x² − 3x − 10
▸ x² − 3x − 10
The linear coefficient is the sum, the constant the product, of the two numbers.
Exam Wrap-up
Three Identities
(a±b)² = a²±2ab+b², (a+b)(a−b) = a²−b²
Foundations of factoring and quadratics
Grade-9 school exam type
Expand (x + 4)(x − 4).
①x² − 8
②x² + 16
③x² − 16
④x² − 8x − 16
⑤x² − 16x
▸ ③ x² − 16
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Use the sum-and-difference product (a + b)(a − b) = a² − b².
(x + 4)(x − 4) = x² − 4²
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Compute.
= x² − 16
🎯 Exam Points
①(a+b)² ≠ a²+b² — never drop 2ab
②Difference of squares appears more in reverse (factoring)