Square of a sum
(a + b)² = a² + 2ab + b².
Use 60² and 5² to get 65²
A square of side 60 has area 60² = 3600. A square of side 5 has area 5² = 25. Can we use them to find 65²?
Which one is 65²?
What this lesson covers
The idea
(a + b)² = a² + 2ab + b²; geometrically, a square of side a + b splits into squares of sides a and b and two rectangles of sides a and b.
Use 60² and 5² to get 65²
A square of side 60 has area 60² = 3600. A square of side 5 has area 5² = 25. Can we use them to find 65²?
Which one is 65²?
- 3600 + 25
- 3600 + 25 + 300
- 3600 + 25 + 600
Fill the empty pieces
The big square has side a + b. It is cut into four pieces. The two squares are in place. Tap each empty piece to fill it. Drag the purple dot to cut at another place.
Two squares and two rectangles
(a + b)² = a² + 2ab + b². A square of side a + b splits into squares of sides a and b and two rectangles of sides a and b.
65² = (60 + 5)² = 60² + 5² + 2 × (60 × 5) = 3600 + 25 + 600 = 4225
It is the four-cell grid from before: (a + b)(a + b) = a × a + a × b + b × a + b × b = a² + 2ab + b² The cells ab and ba are equal, so they join into 2ab. NCERT calls this Identity 1A. If you forget it, multiply the bracket by itself.
Notes
(a + b)² = a² + 2ab + b²: the square of a + b has the square of a, the square of b and two rectangles of area ab.
Check yourself
A square has side 8 + 3. How many squares do the two rectangles together have?
Answer: 48
2ab = 2 × 8 × 3 = 48. Check: (8 + 3)² = 121 = 64 + 48 + 9.
Which is the expansion of (x + 3)²?
Use the identity to find 45². Write 45 as 40 + 5.
Answer: 2025
(40 + 5)² = 1600 + 400 + 25 = 2025.
Expand (6x + 5)². What is the number in front of x?
Answer: 60
(6x + 5)² = (6x)² + 2 × (6x × 5) + 5² = 36x² + 60x + 25.
- x² + 9. That leaves out the two rectangles, 2 × x × 3 = 6x. (x + 3)² is not x² + 3².
- x² + 6x + 9 — correct. Yes! x², the two rectangles 2 × x × 3 = 6x, and 3² = 9.
- x² + 3x + 9. Only one rectangle, 3x, is counted. There are two: 2 × x × 3 = 6x.