Algebraic expressions

4. Algebraic Identities
Exercise 4.1
1. Question

Evaluate each of the following using identities:

(i)

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

(iii)

(iv) (a – 0.1) (a +0.1)

(v)

Answer

(i) We know, (a-b)2 = a2 + b2 -2ab




(ii) We know, (a-b)2 = (a+b) (a-b)

(2x+y) (2x-y) = (2x)2– y2 = 4x2 – y2

(iii) We know, (a-b)2 = a2 + b2 -2ab




(iv) We know, (a-b)2 = (a+b) (a-b)

(a – 0.1) (a +0.1) = (a)2– (0.1)2

= a2 – 0.01

(v) We know, (a-b)2 = a2 - b2




2. Question

Evaluate each of the following using identities:

(i) (399)2

(ii) (0.98)2

(iii) 991×1009

(iv) 117×83

Answer

(i) We will use the identity, (a-b)2 = a2 + b2 – 2ab

,(ii) We will use the identity, (a-b)2 = a2 + b2 – 2ab




(iii) We will use the identity, (a-b)(a+b)= a2 - b2




(iv) We will use the identity, (a-b) (a+b)= a2 - b2




3. Question

Simplify each of the following:

(i) 175 × 175 + 2 × 175 × 25 + 25 ×25

(ii) 322 × 322 – 2 × 322 × 22 + 22 × 22

(iii) 0.76 ×0.76+2×0.76×0.24+0.24×0.24

(iv)

Answer

(i) We know, (a+b)2 = a2 + b2 +2ab




(ii) We know, (a-b)2 = a2 + b2 -2ab




(iii) We know, (a+b)2 = a2 + b2 +2ab




(iv) We know, (a-b)2 = (a-b) (a+b)

,4. Question

If x + = 11, find the value of x2 +

Answer

Here, we will use (a+b)2 = a2 + b2 +2ab




5. Question

If x - = -1, find the value of x2 +

Answer

Here, we will use (a-b)2 = a2 + b2 -2ab




6. Question

If x + = , find the value of x2 + and x4+

Answer

Here, we will use (a+b)2 = a2 + b2 +2ab

, Now,




7. Question

If x2 + = 66, find the value of x -

Answer

Here, we will use (a-b)2 = a2 + b2 -2ab.




8. Question

If x2 + = 79, find the value of x +

Answer

Here, we will use (a+b)2 = a2 + b2 + 2ab




9. Question

If 9x2 +25y2 = 181 and xy = -6, find the value of 3x +5y

Answer

Here, we will use (a+b)2 = a2 + b2 + 2ab