Hello Dear Students Here in this post we have provided for you Algebra Formulas with their Proof by our Expert Math Teacher.
Now days some of students don’t know why (a+b)2 = a2 + 2ab + b2 ; Why (a – b)2 = a2 – 2ab + b2. They direct learn it. And use it. So Here we came for those Students who are don’t the reason behind this Algebra Formulas. Hope this post will help you for deep knowledge in Math.
Square Formulas
Why (a+b)2 = a2 + 2ab + b2
Proof:
(a + b)2 = (a + b) (a + b)
= a2 + a b + a b+ b2
= a2 + 2ab + b2
Why (a – b)2 = a2 – 2ab + b2
Proof:
(a +b)2 = (a – b) (a – b)
= a2 – a b – a b + b2
= a2 – 2ab + b2
Why a2 + b2 = (a + b)2 – 2ab
Proof:
a2 + b2 = a2 + b2 + 2ab – 2ab
= a2 + 2ab + b2 – 2ab
= (a + b)2 – 2ab
Or
a2 + b2 + 2ab – 2ab
= a2 – 2ab + b2 + 2ab
= a2 – 2a + b2 + 2ab
= (a – b)2 + 2ab
Why a2 + b2 = ½ [ (a + b)2 + (a – b)2]
Proof:
a2 + b2 = ½ x 2(a2 + b2)
= ½ x 2 [a2 + b2+ a b +a b]
= ½ [ 2a2+ 2 b2– 2ab]
= ½ [a2 + 2ab + b2 + a2 – 2ab + b2]
= ½ [ (a + b)2 + (a – b)2]
Why a2 – b2 = (a + b) (a- b)
Proof:
a2 – b2 = a2 + a b – a b – b2
= a (a + b) –b (a – b)
= (a + b) (a- b)
Now, Cube Formulas
Why (a + b)3 = = a3 + b3 + 3ab (a +b)
Proof:
(a + b)3 = (a + b)2 (a + b)
= (a2 + 2ab + b2) (a + b)
= a3 + 2a2b + ab2 + a2b + 2ab2 + b3
= a3 + 3a2 b + 3ab2 + b3
= a3 + b3 + 3ab (a +b)
Why (a -b)3 = a3 – b3 – 3ab (a- b)
(a -b)3 = (a – b)2 (a -b)
= (a2 – 2ab + b2) (a -b)
= a3 – 2a2 b + ab2 – a2 b + 2ab2– b3
= a3-b3 – 3a2 b + 3ab2
= a3 – b3 – 3ab (a- b)
We know, (a – b)3 = a3 – b3 – 3ab (a – b)
= a3– b3 = (a -b)3 + 3ab (a – b)
We know, (a + b)3 = a3 + b3 + 3ab (a + b)
= a3 + b3 = (a + b)3 -3ab (a + b)
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Available Soon Sir