01/06/2024
07
গণিতের সূত্র:
1. (𝑎+𝑏)² = 𝑎²+𝑏²+2𝑎𝑏 | = (𝑎 – 𝑏)²+4𝑎𝑏
2. 𝑎² + 𝑏² = (𝑎+𝑏)² – 2𝑎𝑏 | = (𝑎 – 𝑏)²+2𝑎𝑏.
3. 𝑥 + ⅟𝑥 = 𝑛 ➺ 𝑥² + ⅟𝑥² = 𝑛² – 2 | 𝑥³ + ⅟𝑥³ = 𝑛³ – 3𝑛
4. 𝑥 – ⅟𝑥 = 𝑛 ➺ 𝑥² + ⅟𝑥² = 𝑛²+2 | 𝑥³ + ⅟𝑥³ = 𝑛³+ 3𝑛
5. (𝑎 – 𝑏)² = 𝑎² – 2𝑎𝑏+𝑏² | = (𝑎+𝑏)² – 4𝑎𝑏
6. 𝑎² – 𝑏² = (𝑎 +𝑏)(𝑎 – 𝑏)
7. 2(𝑎²+𝑏²) = (𝑎+𝑏)²+(𝑎 – 𝑏)²
8. 4𝑎𝑏 = (𝑎+𝑏)² – (𝑎 – 𝑏)²
9. 𝑎𝑏 = {(𝑎+𝑏)/2}² – {(𝑎 – 𝑏)/2}²
10. (𝑎+𝑏+𝑐)² = 𝑎²+𝑏²+𝑐²+2(𝑎𝑏+𝑏𝑐+𝑐𝑎)
11. (𝑎+𝑏)³ = 𝑎³+3𝑎²𝑏+3𝑎𝑏²+𝑏³
12. (𝑎+𝑏)³ = 𝑎³+𝑏³+3𝑎𝑏(𝑎+𝑏)
13. 𝑎 – 𝑏)³ = 𝑎³ – 3𝑎²𝑏+3𝑎𝑏² – 𝑏³
14. (𝑎 – 𝑏)³ = 𝑎³ – 𝑏³ – 3𝑎𝑏(𝑎 – 𝑏)
15. 𝑎³+𝑏³ = (𝑎+𝑏) (𝑎² – 𝑎𝑏+𝑏²)
16. 𝑎³+𝑏³ = (𝑎+𝑏)³ – 3𝑎𝑏(𝑎+𝑏)
17. 𝑎³ – 𝑏³ = (𝑎 – 𝑏) (𝑎²+𝑎𝑏+𝑏²)
18. 𝑎³ – 𝑏³ = (𝑎 – 𝑏)³+3𝑎𝑏(𝑎 – 𝑏)
19. (𝑎² + 𝑏² + 𝑐²) = (𝑎 + 𝑏 + 𝑐)² – 2(𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎)
20. 2 (𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎) = (𝑎 + 𝑏 + 𝑐)² – (𝑎² + 𝑏² + 𝑐²)
21. (𝑎 + 𝑏 + 𝑐)³ = 𝑎³ + 𝑏³ + 𝑐³ + 3 (𝑎 + 𝑏) (𝑏 + 𝑐) (𝑐 + 𝑎)
22. 𝑎³ + 𝑏³ + 𝑐³ – 3𝑎𝑏𝑐 = (𝑎+𝑏+𝑐)(𝑎² + 𝑏²+ 𝑐²–𝑎𝑏–𝑏𝑐– 𝑐𝑎)
23. 𝑎3 + 𝑏3 + 𝑐3 – 3𝑎𝑏𝑐 = ½ (𝑎+𝑏+𝑐) { (𝑎–𝑏)²+(𝑏–𝑐)²+(𝑐–𝑎)²}
24. (𝑥 + 𝑎) (𝑥 + 𝑏) = 𝑥² + (𝑎 + 𝑏) 𝑥 + 𝑎𝑏
25. (𝑥 + 𝑎) (𝑥 – 𝑏) = 𝑥² + (𝑎 – 𝑏) 𝑥 – 𝑎𝑏
26. (𝑥 – 𝑎) (𝑥 + 𝑏) = 𝑥² + (𝑏 – 𝑎) 𝑥 – 𝑎𝑏
27. (𝑥 – 𝑎) (𝑥 – 𝑏) = 𝑥² – (𝑎 + 𝑏) 𝑥 + 𝑎𝑏
28.(𝑥+p) (𝑥+q) (𝑥+r) = 𝑥³ + (p+q+r) 𝑥² + (pq+qr+rp) 𝑥 +pqr
29. 𝑏𝑐 (𝑏 – 𝑐) + 𝑐𝑎(𝑐 – 𝑎) + 𝑎𝑏(𝑎 – 𝑏) = – (𝑏 – 𝑐) (𝑐 – 𝑎) (𝑎 – 𝑏)
30. 𝑎² (𝑏 – 𝑐) + 𝑏²(𝑐 – 𝑎) + 𝑐²(𝑎 – 𝑏) = – (𝑏 – 𝑐) (𝑐 – 𝑎) (𝑎 – 𝑏)
31. 𝑎 (𝑏² – 𝑐²) + 𝑏(𝑐² – 𝑎²) + 𝑐(𝑎² – 𝑏²) = (𝑏 – 𝑐) (𝑐 – 𝑎) (𝑎 – 𝑏)
32. 𝑎³ (𝑏 – 𝑐) + 𝑏³ (𝑐 – 𝑎) +𝑐³ (𝑎 – 𝑏) = – (𝑏 – 𝑐) (𝑐 – 𝑎) (𝑎 – 𝑏)(𝑎 + 𝑏 + 𝑐)
33. 𝑏² – 𝑐²(𝑏² – 𝑐²) + 𝑐²𝑎²(𝑐² – 𝑎²)+𝑎²𝑏²(𝑎² – 𝑏²) = – (𝑏 – 𝑐) (𝑐 – 𝑎) (𝑎 – 𝑏) (𝑏+𝑐) (𝑐+𝑎) (𝑎+𝑏)
34. (𝑎𝑏 + 𝑏𝑐+𝑐𝑎) (𝑎+𝑏+𝑐) – 𝑎𝑏𝑐 = (𝑎 + 𝑏)(𝑏 + 𝑐) (𝑐+𝑎)
35. (𝑏 + 𝑐)(𝑐 + 𝑎)(𝑎