Expansions

We shall discuss expansions of binomials and trinomials

Formulae

Following formulae are got by multiplying out the brackets.

e.g., (a + b)² = (a + b)(a + b)

= a(a + b) + b(a + b)

= a² + ab + ba + b²

= a² + 2ab + b²

(1) (a + b)² = a² + 2ab + b²

(2) (a - b)² = a² - 2ab + b²

(3) (a + b) (a - b) = a² - b²

(4) (i) (x + a) (x + b) = x² + (a + b)x + ab

(ii) (x + a) (x - b) = x² + (a - b)x - ab

(iii) (x - a) (x - b) = x² + (- a - b)x + ab

In R.H.S of (1) and (2), the middle term

i.e.,

= 4 T₁ T₃

and

Expand the following:

(i) (3a + 4b)² (ii) (x - 5y)²

(i) (3a + 4b)² = (3a)² + 2 (3a) x (4b) + (4b)²

= 9a² + 24ab + 16b²

(ii) (x - 5y)² = (x)² - 2 (x) (5y) + (5y)²

= x² - 10xy + 25y²

Find algebraically the value of 205².

205² = (200 + 5)²

= (200)² + 2 (200) (5) + (5)²

= 40000 + 2000 + 25

= 42025

If the expression 36x² + Kx + 25 is a perfect square, find K.

Middle term of the given expression

= 2 6x 5

= 60 x

K = 60

More Formulae

1. a² + b² = (a + b)² - 2ab

2. a² + b² = (a - b)² + 2ab

5. (a + b)² = (a - b)² + 4ab

7. (a + b + c)² = a² + b² + c² + 2 (ab + bc + ca)

8. (a + b)³ = a³ + 3a²b + 3ab² + b³

= a³ + b³ + 3ab (a + b)

9. (a - b)³ = a³ - 3a²b + 3ab² - b³

= a³ - b³ - 3ab (a - b)

If x + y = 10 and xy = 21, find x² + y².

x² + y² = (x + y)² - 2xy

= (10)² - 2 21

= 100 - 42 = 58

Expand: (a + 3b - 4c)²

(a + 3b - 4c)² = (a)² + (3b)² + (-4c)² + 2[(a) (3b) + (3b) (-4c)

+ (-4c) (a)]

= a² + 9b² + 16c² + 6ab - 24bc - 8ac

If x² + y² + z² = 38, x + y + z = 10, find xy + yz + zx.

We have (x² + y² + z²) + 2 (xy + yz + zx) = (x + y + z)²

38 + 2 (xy + yz + zx) = (10)²

2(xy + yz + zx) = 100 - 38

= 62

xy + yz + zx = 31

Expand: (5a - 2b)³

(5a - 2b)³ = (5a)³ - 3(5a)² (2b) + 3(5a) (2b)² - (2b)³

= 125a³ - 150a²b + 60ab² - 8b³

If find

64 + 12 =