Class 8 Identities വർഗസമവാക്യങ്ങൾ
In Class 8 you met three famous identities about squares. Chapter 4 uses them in one crucial place: turning a product and a sum of two numbers into their difference, and then into the numbers themselves. This page revises the identities with pictures and practises that step.
1. (x + y)² = x² + 2xy + y²
A square of side x + y can be cut into four pieces: a square x², a square y², and two rectangles xy.
2. (x − y)² = x² − 2xy + y²
Start with a square of side x. Cut a strip of width y from the right and from the bottom; the corner piece y² is removed twice, so it must be added back once.
3. (x + y)(x − y) = x² − y²
Take a square x² and remove a small square y² from a corner. The L-shape that remains can be cut and rearranged into a rectangle (x + y) by (x − y).
(x − y)² = x² − 2xy + y²
(x + y)(x − y) = x² − y²
Read x² − y² = (x + y)(x − y) backwards too: a difference of two squares factorises into (sum)(difference). The chapter uses it as x² − (8 − x)² = (x + 8 − x)(x − 8 + x).
4. The bridge: (x − y)² = (x + y)² − 4xy
Subtract the second identity from the first: (x + y)² − (x − y)² = 4xy. Rearranged, this gives two very useful forms:
(x − y)² = (x + y)² − 4xy(x + y)² = (x − y)² + 4xy
5. Quick squares and products
The identities turn hard products into easy ones by using a nearby round number:
- 101² = (100 + 1)² = 10000 + 200 + 1 = 10201
- 99² = (100 − 1)² = 10000 − 200 + 1 = 9801
- 51 × 49 = (50 + 1)(50 − 1) = 2500 − 1 = 2499
- 35² = (30 + 5)² = 900 + 300 + 25 = 1225
6. From sum and difference to the numbers സമവാക്യജോടികൾ
If two numbers add to 44 and differ by 4, the bigger is half of (44 + 4) = 24 and the smaller is half of (44 − 4) = 20. In general:
x = ½(sum + difference)y = ½(sum − difference)
Why it works: x + y plus x − y is 2x, so x is half the total; x + y minus x − y is 2y. (Sum and difference are both even or both odd, which is why the sliders move in steps of 2.)
7. From product and sum to the numbers
This is the step the chapter uses. Suppose xy = 480 and x + y = 44. The bridge identity gives the difference, and then the two numbers.
Not every pair of values is possible: (x + y)² − 4xy must be a perfect square (or at least not negative). Try sum 70 with product 1200, or sum 30 with product 221.
A farmer knows a rectangular plot has 800 sq m of area and needs 120 m of fencing (so length + breadth = 60). The bridge identity gives (length − breadth)² = 60² − 4 × 800 = 400, so the sides differ by 20: the plot is 40 m by 20 m. Exactly this appears in Exercise 4 of the chapter.
8. Check yourself
- (x + y)² = x² + 2xy + y²; (x − y)² = x² − 2xy + y²; (x + y)(x − y) = x² − y².
- Bridge: (x − y)² = (x + y)² − 4xy and (x + y)² = (x − y)² + 4xy.
- Sum and difference known ⇒ x = ½(s + d), y = ½(s − d).
- Product and sum known ⇒ find (x − y)² from the bridge, take the square root, then use the line above.