Question

If 4x²+1/x²=12, then find 8x²+1/x³.​

Answer 1

Step-by-step explanation:

4x^2+1/x^2=12

(2x)^2+(1/x)^2+2×2x×1/x=12+4

(2x+1/x)^2=16

2x+1/x=+/-4……………(1)

cubing both side

(2x)^3+(1/x)^3+3×2x×1/x(2x+1/x)=(+/-4)^3

8x^3+1/x^3+6×(+/-4)=+/-64

8x^3+1/x^3=+/-64 - (+/-24) =+/-40 , Answer

Answer 2

Step-by-step explanation:

Given:-

4x^2 + 1/x^2 = 12

To find out:-

Values of 8x^3 + 1/x^3

Solution:-

We have

4x^2 + 1/x^2 = 12

This express can be also written as in the form of (a+b)^2 = a^2 + 2ab + b^2

Where we have to put a = 2x and b = 1/x, we get

→ (2x + 1/x)^2 = 12

→ (2x)^2 + 2(2x)(1/x) + (1/x)^2 = 12

→ 4x^2 + 2(2) + (1/x)^2 = 12

→ 4x^2 + 4 + 1/x^2 = 12

→ 4x^2 + 1/x^2 = 12 - 4

→ 4x^2 + 1/x^2 = 9

∴ (2x + 1/x)^2 = 9

→ 2x + 1/x = √9

→ 2x + 1/x = 3

Now, cubing on both sides, we get

(2x + 1/x)^3 = (3)^3

[∵ (a+b)^3 = a^3 + b^3 + 3ab(a+b), Where we have to put in our expression a = 2x and b = 1/x]

→ (2x)^2 + (1/x)^3 + 3(2x)(1/x)(2x + 1/x) = (3)^3

→ 4x^3 + 1/x^3 + 6(2x + 1/x) = 27

→ 4x^3 + 1/x^3 + 6(3) = 27

→ 4x^3 + 1/x^3 + 18 = 27

→ 4x^3 + 1/x^3 = 27 - 18

→ 4x^3 + 1/x^3 = 9