In the expansion of (x^2-b/x)^4, the coefficient of x^2 is 294. Find the value of the constant "b"
Using the binomial expansion
(x^2-b/x)^4
= (x^2)^4 + 4(x^2)^3 (-b/x) + 6(x^2)^2 (-b/x)^2 + 4(x^2) (-b/x)^3 + (-b/x)^4
= 4x^8 - 4x^6 (b/x) + 6x^4 b^2/x^2 - ...
= 4x^8 - 4x^5 b + 6x^2 b^2 - ...
so the term containing x^2 is 6x^2 b^2 , which is equal to 294
6v^2 = 294
b^2 = 49
b = ± 7
check by Wolfram:
http://www.wolframalpha.com/input/?i=expand+(x%5E2-7%2Fx)%5E4
http://www.wolframalpha.com/input/?i=expand+(x%5E2%2B7%2Fx)%5E4
To find the value of the constant "b," we need to examine the coefficient of the x^2 term in the expansion:
(x^2 - b/x)^4 = C4,0(x^2)^4*(-b/x)^0 + C4,1(x^2)^3*(-b/x)^1 + C4,2(x^2)^2*(-b/x)^2 + C4,3(x^2)^1*(-b/x)^3 + C4,4(x^2)^0*(-b/x)^4
Here, C4,0, C4,1, C4,2, C4,3, C4,4 represent the binomial coefficients for corresponding terms.
The x^2 term only arises in one of the terms:
C4,2(x^2)^2*(-b/x)^2 = 294
Simplifying this term:
C4,2 * x^4 * b^2/x^2 = 294
C4,2 is the binomial coefficient for choosing 2 objects from 4, which is calculated as:
C4,2 = 4! / (2! * (4 - 2)!) = 6
Substituting this value:
6 * x^4 * b^2/x^2 = 294
Simplifying further:
6 * x^2 * b^2 = 294
At this point, we can isolate "b" by dividing both sides of the equation by 6 * x^2:
b^2 = 294 / (6 * x^2)
b^2 = 49 / x^2
To find the value of "b," we need to know the value of "x." If you provide the value of "x," we can find the corresponding value of "b."