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| Expansions |
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| We shall discuss expansions of binomials and trinomials |
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| Following formulae are got by multiplying out the brackets. |
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| e.g., (a + b)2 = (a + b)(a + b) |
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| = a(a + b) + b(a + b) |
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| = a2 + ab + ba + b2 |
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| = a2 + 2ab + b2 |
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| (1) (a + b)2 = a2 + 2ab + b2 |
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| (2) (a - b)2 = a2 - 2ab + b2 |
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| (3) (a + b) (a - b) = a2 - b2 |
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| (4) (i) (x + a) (x + b) = x2 + (a + b)x + ab |
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| (ii) (x + a) (x - b) = x2 + (a - b)x - ab |
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| (iii) (x - a) (x - b) = x2 + (- a - b)x + ab |
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| In R.H.S of (1) and (2), the middle term |
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i.e.,  |
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 = 4  T1  T3 |
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 and  |
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| Expand the following: |
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| (i) (3a + 4b)2 (ii) (x - 5y)2 |
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(i) (3a + 4b)2 = (3a)2 + 2  (3a) x (4b) + (4b)2 |
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| = 9a2 + 24ab + 16b2 |
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(ii) (x - 5y)2 = (x)2 - 2  (x)  (5y) + (5y)2 |
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| = x2 - 10xy + 25y2 |
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| Find algebraically the value of 2052. |
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| 2052 = (200 + 5)2 |
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= (200)2 + 2  (200) (5) + (5)2 |
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| = 40000 + 2000 + 25 |
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| = 42025 |
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| If the expression 36x2 + Kx + 25 is a perfect square, find K. |
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| Middle term of the given expression |
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= 2  6x  5 |
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| = 60 x |
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 K = 60 |
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| 1. a2 + b2 = (a + b)2 - 2ab |
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| 2. a2 + b2 = (a - b)2 + 2ab |
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| 5. (a + b)2 = (a - b)2 + 4ab |
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| 7. (a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca) |
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| 8. (a + b)3 = a3 + 3a2b + 3ab2 + b3 |
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| = a3 + b3 + 3ab (a + b) |
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| 9. (a - b)3 = a3 - 3a2b + 3ab2 - b3 |
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| = a3 - b3 - 3ab (a - b) |
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| If x + y = 10 and xy = 21, find x2 + y2. |
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| x2 + y2 = (x + y)2 - 2xy |
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= (10)2 - 2  21 |
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| = 100 - 42 = 58 |
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| Expand: (a + 3b - 4c)2 |
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| (a + 3b - 4c)2 = (a)2 + (3b)2 + (-4c)2 + 2[(a) (3b) + (3b) (-4c) |
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| + (-4c) (a)] |
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| = a2 + 9b2 + 16c2 + 6ab - 24bc - 8ac |
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| If x2 + y2 + z2 = 38, x + y + z = 10, find xy + yz + zx. |
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| We have (x2 + y2 + z2) + 2 (xy + yz + zx) = (x + y + z)2 |
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| 38 + 2 (xy + yz + zx) = (10)2 |
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| 2(xy + yz + zx) = 100 - 38 |
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| = 62 |
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| xy + yz + zx = 31 |
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| Expand: (5a - 2b)3 |
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| (5a - 2b)3 = (5a)3 - 3(5a)2 (2b) + 3(5a) (2b)2 - (2b)3 |
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| = 125a3 - 150a2b + 60ab2 - 8b3 |
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If  find  |
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64 + 12 =  |
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