Perfect Square Identities

1Summary 重點摘要

Learn two multiplication formulas: (a±b)² = a² ± 2ab + b².

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Two formulasThere are two perfect-square identities, to be memorised exactly: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². Their structure is identical: squares at both ends and twice the product, 2ab, in the middle; the only difference is the sign of the middle term — a plus inside the bracket gives a positive middle term, a minus gives a negative one. The end terms are always positive, since no square is negative. These two identities are cornerstones of secondary algebra, serving expansion, factorisation and later the solving of quadratic equations, so they are worth drilling until fast and accurate.
2. Step 2: Why it worksThe identities can be verified by long multiplication, and knowing the source makes them harder to misuse. (a + b)² means (a + b)(a + b): multiplying term by term gives a×a + a×b + b×a + b×b = a² + ab + ab + b², and the two identical ab terms merge into 2ab, so (a + b)² = a² + 2ab + b². Likewise (a − b)² = (a − b)(a − b) = a² − ab − ab + b² = a² − 2ab + b², where the end term b² stays positive because the two negatives multiply to a positive. Check with numbers: with a = 2 and b = 3, (2 + 3)² = 25 while 2² + 2(2)(3) + 3² = 4 + 12 + 9 = 25 — equal, as promised.
3. Step 3: Using it to expandOn seeing the form (a + b)² or (a − b)², apply the identity directly — far faster than term-by-term multiplication. For (x + 3)²: match a = x and b = 3, giving x² + 2(x)(3) + 3² = x² + 6x + 9. Two more: (y − 5)² = y² − 2(y)(5) + 5² = y² − 10y + 25; and (2x + 1)² = (2x)² + 2(2x)(1) + 1² = 4x² + 4x + 1, where a itself being 2x means its square is (2x)² = 4x², never 2x². The three examples display the common shapes — plain variable, minus sign, leading coefficient — and with practice each is recognised at a glance.
4. Step 4: Using it to factoriseUsed backwards, the identities become factorisation tools: a² + 2ab + b² becomes (a + b)², and a² − 2ab + b² becomes (a − b)². Take x² + 6x + 9: check three conditions — the head x² is a square (x squared), the tail 9 is a square (3²), and the middle 6x equals 2 × x × 3 = 6x; all three hold, so x² + 6x + 9 = (x + 3)². A minus version: x² − 8x + 16 = x² − 2(x)(4) + 4² = (x − 4)². The screening chant: first verify that head and tail are squares, then verify that the middle is twice the product; only after passing both tests may the identity be applied.
5. Step 5: Common mistakesThe classic error with perfect squares is dropping the middle term, assuming (a + b)² = a² + b². Numbers expose it at once: with a = 2 and b = 3, (2 + 3)² = 25 but 2² + 3² = 13, a gap of exactly the missing 2ab. The correct expansion is (x + 3)² = x² + 6x + 9, never x² + 9. A second error arises squaring coefficients: (2x)² = 4x², not 2x². A third concerns the sign: (x − 3)² = x² − 6x + 9, with a negative middle but a still-positive end; writing x² − 6x − 9 is wrong. After expanding, count the terms: a perfect square always unfolds into three.
6. Step 6: Numerical checkingTesting identities with concrete numbers is a protective habit. Substitute a small number (such as 1 or 2) into both the original and the expanded form; the two must agree. For (x + 3)² = x² + 6x + 9, put x = 1: the left side is (1 + 3)² = 16 and the right side is 1 + 6 + 9 = 16 — equal, so the expansion stands. Had the right side been wrongly written as x² + 9, it would give 10 against 16, instantly exposing the error. This technique, numerical checking, shines in the last minutes of an exam: thirty seconds of substitution can rescue the whole question. Agreement at two different numbers makes correctness very likely.
7. Step 7: The chant and memory aidsA Chinese chant helps: 'square the first, square the last, twice the product in the middle' — square the first term for the opening, square the last for the closing, multiply the two and double for the middle; the sign matches the bracket. Pair it with structural memory: a perfect square always expands to three terms, with the sign pattern 'positive, same as the bracket, positive'. Geometry cements it: a square of side (a + b) cuts into an a² square, a b² square and two ab rectangles, whose areas total a² + 2ab + b² — the identity itself. Linking the algebra to a picture makes the memory firmer and shows the formula is no arbitrary rule but a necessary fact about area.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• (a+b)² = a² + 2ab + b²
• (a-b)² = a² - 2ab + b²
• Must have middle term 2ab
• b² is always positive
• Chant: first square, last square, twice product in middle