Problem 1 :
Expand (P - 2T)2
Solution :
Here a = P and b = 2T
(P - 2 T)2 = P2 - 2 (P)(2T) + (2T)2
= P2 - 4 PT + 4T2
Problem 2 :
Expand (2P - Q)²
Solution :
Here a = 2P and b = Q
(2 P - Q)2 = (2P)2 - 2 (2P)(Q) + Q2
= 4P2 - 4PQ + Q2
Problem 3 :
Expand (2a - 3b)²
Solution :
here a = 2a and b = 3b
(2a - 3b)2 = (2a)2 - 2 (2a)(3b) + (3b)2
= 4a2 - 12ab + 9b2
Problem 4 :
Expand (5N - 3P)²
Solution :
here a = 5N and b = 3P
(2a - 3b)2 = (5N)2 - 2(5N)(3P) + (3P)2
= 25N2 - 30NP + 9P2
Problem 5 :
Expand (9T - Q)²
Solution :
here a = 9T and b = Q
(9T - Q)2 = (9T)2 - 2 (9T)(Q) + Q2
= 81T2 - 18 TQ + Q2
Problem 6 :
Expand (13p - 5q)²
Solution :
Here a = 13 P and b = 5 Q
(13P - 5Q)2 = (13P)2 - 2 (13P)(5Q) + (5Q)2
= 169P2 - 130 PQ + 25 Q 2
Problem 7 :
Expand (3p - 4q)²
Solution :
Here a = 3p and b = 4q
(3p - 4q)2 = (3p)2 - 2(3p)(4q) + (4q)2
= 9P2 - 24pq + 16q 2
Problem 8 :
x2 + 1/x2 = 47, then the value of x + 1/x
Solution :
Using the formula (a + b)2 = a2 + 2ab + b2
a2 + b2 = (a + b)2 - 2ab
x2 + 1/x2 = (x + 1/x)2 - 2x(1/x)
47 = (x + 1/x)2 - 2
(x + 1/x)2 = 47 + 2
(x + 1/x)2 = 49
x + 1/x = √49
x + 1/x = -7 and 7
Problem 9 :
x2 + 1/x2 = 66, then the value of x - 1/x
Solution :
Using the formula (a - b)2 = a2 - 2ab + b2
a2 + b2 = (a - b)2 + 2ab
x2 + 1/x2 = (x - 1/x)2 + 2x(1/x)
66 = (x - 1/x)2 + 2
(x - 1/x)2 = 66 - 2
(x - 1/x)2 = 64
x - 1/x = √64
x - 1/x = -8 and 8
Problem 10 :
Using the identity for the square of a binomial, evaluate the following
(0.98)2
Solution :
(0.98)2 = (1 - 0.02)2
= (1 - 2/100)2
Writing the expansion of (a - b)2
= 12 - 2(1)(2/100) + (2/100)2
= 1 - (4/100) + (4/10000)
= 1 - 0.04 + 0.0004
= 1.0004 - 0.04
= 0.9604
Problem 11 :
Find the square of (3x/4 - 4y/5)
Solution :
(3x/4 - 4y/5)2
Here a = 3x/4 and b = 4y/5
= (3x/4)2 - 2(3x/4)(4y/5) + (4y/5)2
= (9x2/16) - (24xy/20) + (16y2/25)
= (9x2/16) - (6xy/5) + (16y2/25)
Problem 12 :
236 x 236 - 2 (236)(86) + 86 x 86
Solution :
236 x 236 - 2 (236)(86) + 86 x 86
= (236 - 86)2
= 1502
= 22500
Problem 13 :
1.06 x 1.06 - 2 (1.06)(0.06) + 0.06 x 0.06
Solution :
1.06 x 1.06 - 2 (1.06)(0.06) + 0.06 x 0.06
= (1.06 - 0.06)2
= 12
= 1
Problem 14 :
287 x 287 + 269 x 269 - 2 x 287 x 269
Solution :
= 287 x 287 + 269 x 269 - 2 x 287 x 269
= 2872 + 2692 - 2 (287)(269)
= (287 - 269)2
= 182
= 324
Problem 14 :
If a - b = 3 and a2 + b2 = 29, find the value of ab
Solution :
Given that, a - b = 3 and a2 + b2 = 29
a2 + b2 = (a - b)2 + 2ab
Applying these values in the formula, we get
29 = 32 + 2ab
29 - 9 = 2ab
2ab = 20
ab = 20/2
ab = 10
So, the value of ab is 10.
Problem 15 :
Expand and simplify (x + 3)2 + (x - 4)2
Solution :
= (x + 3)2 + (x - 4)2
= x2 + 2x(3) + 32 + x2 - 2x(4) + 42
= x2 + 6x + 9 + x2 - 8x + 16
= 2x2 - 2x + 25
Problem 15 :
Rationalize the denominator 1/(√9 - √8)
Solution :
= 1/(√9 - √8)
To rationalize the denominator, we should multiply both numerator and denominator by the conjugate of the denominator.
= [1/(√9 - √8)] [(√9 + √8)/(√9 - √8)]
= [(√9 + √8)/(√9 + √8)(√9 - √8)]
= (√9 + √8)/(√92 - √82)