if f(x) ax^2+bx+c , g(x) :1/3x^2+2 and fg(x)= 18x^4+24x^2+11/(3x^2+2)^2. find a ,b and c
Well, thinking fg(x) means f(x) * g(x) I get no consistent solution. So, making
fg(x) = f(g(x)) we have
f(g) = ag^2 + bg + c
= a/(3x^2+2)^2 + b/(3x^2+2) + c
= [a + b(3x^2+2) + c(3x^2+2)^2] / (3x^2+2)^2
= 9cx^4 + (12c+3b)x^2 + (4c+2b+a)
= 18x^4 + 24x^2 + 11
That makes c=2, b=0, a=3, so f(x) = 2x^2 + 3
solve it in a way am goin 2 understand
To find the values of a, b, and c in the function f(x) = ax^2 + bx + c, we need to consider the product of f(x) and g(x), and then compare it to the given expression fg(x) = 18x^4 + 24x^2 + 11 / (3x^2 + 2)^2.
Let's multiply f(x) and g(x) to find the expression fg(x):
f(x) * g(x) = (ax^2 + bx + c) * (1/3x^2 + 2)
= (1/3) * ax^2 * x^2 + (1/3) * bx * x^2 + (1/3) * cx^2 + 2ax^2 + 2bx + 2c
= (a/3)x^4 + (b/3)x^3 + (c/3)x^2 + 2ax^2 + 2bx + 2c
= (a/3)x^4 + (b/3)x^3 + (5a/3 +c )x^2 + 2bx + 2c
Comparing this with the given expression fg(x) = 18x^4 + 24x^2 + 11 / (3x^2 + 2)^2,
we can equate the coefficients of corresponding powers of x:
For x^4: (a/3) = 18 => a = 54
For x^3: (b/3) = 0 => b = 0
For x^2: (5a/3 + c) = 24 => (5*54/3 + c) = 24 => 90 + c = 24 => c = -66
Therefore, the values of a, b, and c are a = 54, b = 0, and c = -66.