Mathematicals formula and its uses

.Some Useful Formula for All Classes..

Written by:

Mohammod Tofazzal Hossan,

Magura, Panchagarh

1. (a+b)2= a2+2ab+b2 Proof: (a+b)2 =(a+b)(a+b) =a(a+b)+b(a+b) =a.a+a.b+a.b+b.b =a2+2ab+b2 Example:(2x+3y)2 =(2x)2+2.2x.3y+(3y)2 =4x2+12xy+9y2 Ans.	2. (a-b)2= a2-2ab+b2 Proof: (a-b)2 =(a-b)(a-b) =a(a-b)-b(a-b) =a.a-a.b-a.b+b.b =a2-2ab+b2 Example:(3x-4y)2 =(3x)2-2.3x.4y+(4y)2 =9x2-24xy+16y2Ans.
3. a2+b2= (a+b)2-2ab Proof:(a+b)2-2ab =(a+b)(a+b)-2ab =a.a-a.b-a.b+b.b-2ab =a2-2ab+b2-2ab = a2+b2	4. a2-b2= (a+b)(a-b) Proof:(a+b)(a-b) =a(a-b)+b(a-b) =a.a-a.b+a.b-b.b =a2-ab+ab-b2-2ab = a2-b2
5. 2(a2+b2)= (a+b)2+(a-b)2 Proof:(a+b)2+(a-b)2 =(a+b)(a+b)+(a-b)(a-b) =a.a+a.b+a.b+b.b+a.a-a.b-a.b+b.b =a2+2ab+b2+a2-2ab+b2 =2a2+2b2 =2(a2+b2)	6. (a+b)2= (a-b)2+4ab Proof: (a-b)2+4ab =a2-2ab+b2+4ab =a2+2ab+b2 =(a+b)2
7. (a+b)3= a3+3a2b+3ab2+b3 Proof: (a+b)3 =(a+b)(a+b)(a+b) =(a+b){a(a+b)+b(a+b)} =(a+b)(a.a+a.b+a.b+b.b) =a(a.a+a.b+a.b+b.b)+b(a.a+a.b+a.b+b.b) =a.a.a+a.a.b+a.a.b+a.b.b+a.a.b+a.b.b+a.b.b+b.b.b =a3+a2b+a2b+ab2+a2b+ab2+ab2+b3 =a3+3a2b+3ab2+b3	8. (a-b)3= a3-3a2b+3ab2-b3 Proof: (a-b)3 =(a-b)(a-b)(a-b) =(a-b){a(a-b)-b(a-b)} =(a-b)(a.a-a.b-a.b+b.b) =a(a.a-a.b-a.b+b.b)-b(a.a-a.b-a.b+b.b) =a.a.a-a.a.b-a.a.b+a.b.b-a.a.b+a.b.b+a.b.b-b.b.b =a3-a2b-a2b+ab2-a2b+ab2+ab2-b3 =a3-3a2b+3ab2-b3
9. a3+b3=(a+b)3-3ab(a+b) =(a+b)(a2-ab+b2) Proof: (a+b)3-3ab(a+b) =a3+3a2b+3ab2+b3-3ab.a-3ab.b =a3+3a2b+3ab2+b3-3a2b-3ab2 = a3+b3 Again: (a+b)(a2-ab+b2) =a(a2-ab+b2)+b(a2-ab+b2) =a.a2-a.ab+a.b2+b.a2-b.ab+b.b2 =a3-a2b+ab2+ba2-ab2+b3 =a3+b3	10. a3-b3=(a-b)3+3ab(a-b) =(a-b)(a2+ab+b2) Proof: (a-b)3+3ab(a-b) =a3-3a2b+3ab2-b3+3ab.a-3ab.b =a3-3a2b+3ab2-b3+3a2b-3ab2 = a3-b3 Again: (a-b)(a2-ab+b2) =a(a2+ab+b2)-b(a2+ab+b2) =a.a2+a.ab+a.b2-b.a2-b.ab-b.b2 =a3+a2b+ab2-ba2-ab2-b3 =a3-b3

Mathematical Problems solved:

Problem-1. Analyze the product: (a3+8) Solved: (a3 + 8) = (a)3 + (2)3 = ( a + 2){(a)2 - a .2 + (2) 2} =(a + b)(a2 - 2a + 4) Ans: ( a + b)(a2 - 2a + 4)	Problem-2. Analyze the product: (8x3 + 343) Solved: (8x3 + 343) = (2x)3 + (7) 3 = (2x + 7){(2x)2 - 2x . 7 + (7) 2} = (2x + 7)(4x2 - 14x + 49) Ans: (2x + 7)(4x2 - 14x + 49)
Problem-3. Analyze the product: (8a4 + 27ab3) Solved: (8a4 + 27ab3) =a(8a3 + 27b3)[ a এর কারণে সূত্র প্রয়োগ করা যায় না] =a{ (2a)3 + (3b) 3} =a(2a + 3b){(2a)2 - 2a . 3b + (3b) 2} = a(2a + 3b)(4a2 - 6ab + 9b2) Ans: a(2a + 3b)(4a2 - 6ab + 9b2)	Problem-4. Analyze the product: (8x3 +1) Solved: (8x3 + 1) =(8x3 + 13) = (2a)3 + (1) 3} =(2x + 1){(2x)2 - 2x . 1 + (1) 2} = (2x + 1)(4x2 - 2x + 1) Ans: (2x + 1)(4a2 - 2x + 1)
Problem-5. Analyze the product: (64a3 - 125b3) Solved: (64a3 - 125b3) =(4a)3 - (5b) 3 =(4a - 5b){(4a)2 + 4a . 5b + (5b) 2} = (4a - 5b)(16a2 + 20ab + 25b2) Ans: (4a - 5b)(16a2 + 20ab + 25b2)	Problem-6. Analyze the product: (56x3 - 189y3) Solved: (56x3 - 189y3) =7(8x3 - 27y3)[ 7 এর কারণে সূত্র প্রয়োগ করা যায় না] =7{(2x)3 - (3y) 3} =7(2x - 3y){(2x)2 + 2x . 3y + (3y) 2} = 7(2x - 3y)(4x2 + 6xy + 9y2) Ans: 7(2x - 3y)(4x2 + 6xy + 9y2)
Problem-7. Analyze the product: (a2 - 2ab + b2 - p2) Solved: (a2 - 2ab + b2 - p2) =(a)2 - 2.a.b + (b)2 - (p)2) =(a - b)2 - (p)2) [ বর্গমূল করা হয়েছে ] ={(a - b) + (p)}{(a - b) - (p)} [ a2 - b2 সূত্র প্রয়োগ করা হয়েছে ] =(a - b + p)(a - b - p) Ans: =(a - b + p)(a - b - p)	Problem-8. Analyze the product: (a2 - 2ab + 2b - 1) Solved: (a2 - 2ab + 2b - 1) = (a2- 1 - 2ab + 2b ) = (a2- 12 - 2b(a - 1) = (a + 1)(a - 1) - 2b(a - 1) = (a - 1)(a + 1 - 2b) [ সাধারণ গুণনীয়ক নেয়া হয়েছে ] = (a - 1)(a - 2b + 1 ) Ans: (a - 1)(a - 2b + 1 )
Problem-9. Analyze the product: (x4 - 2x2 + 1) Solved: (x4 - 2x2 + 1) = (x2)2 - 2 . x2 . 1 + 12) = (x2 - 1)2 = {(x + 1)(x - 1)}2 Ans: {(x + 1)(x - 1)}2

Problem-9. Analyze the product: (x⁴ - 2x² + 1)
Solved:
(x⁴ - 2x² + 1)
= (x²)² - 2 . x² . 1 + 1²)
= (x² - 1)²
= {(x + 1)(x - 1)}²
Ans: {(x + 1)(x - 1)}²
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