Given that f(x) = 5x^2-3x+7 and f(g(x)) = (5x^4)/9 + (17x^2)/3 + 21, find all possible values for the sum of the coefficients in the quadratic function g(x).
well, the brute force method goes like this:
f(g(x)) = 5g^2-3g+7
If g(x) = ax^2+bx+c, that means
f(g(x)) = 5(ax^2+bx+c)^2 + 3(ax^2+bx+c) + 7
= (5a^2)x^4 + (10ab)x^3 + (10ac+3a+5b^2)x^2 + (10bc+3b)x + (5c^2+3c+7)
For that to be the same as the given f(g(x)) the coefficients must match, so
5a^2 = 5/9
10ab=0
10ac+3a+5b^2 = 17/3
10bc+3b = 0
5c^2+3c+7 = 21
Clearly, a = ±1/3 and b=0
And it is easy to see that
(a,b,c) = (1/3, 0, -2) or (-1/3, 0, 7/5)
I'll have to ponder on whether there is a more direct method.
Why did the quadratic function cross the road? To find all possible values for the sum of the coefficients, of course!
Now, let's clown around with some math. We know that f(g(x)) is equal to (5x^4)/9 + (17x^2)/3 + 21. So, we need to figure out what g(x) looks like.
Let g(x) = ax^2 + bx + c. Now, let's plug g(x) into f(x) and see what happens:
f(g(x)) = f(ax^2 + bx + c)
After some mathematical clowning around, we get:
f(g(x)) = 5(ax^2 + bx + c)^2 - 3(ax^2 + bx + c) + 7
Now, we need to expand and simplify this expression to match the given expression:
f(g(x)) = 5a^2x^4 + (10ab + 5ac - 3a)x^2 + (5b^2 + 10bc - 3b)x + (5c^2 - 3c + 7)
Comparing coefficients, we can solve the following system of equations:
5a^2 = 5/9 (1)
10ab + 5ac - 3a = 17/3 (2)
5b^2 + 10bc - 3b = 0 (3)
5c^2 - 3c + 7 = 21 (4)
Now, let's solve this circus of equations!
From equation (1), we have a^2 = 1/9, which means a could be -1/3 or 1/3. This leads to two different possibilities for g(x).
If a = -1/3, we can substitute it back into equations (2) and (3) to get:
-10b/3 - 5c/3 + 1/3 = 17/9 (5)
5b^2 + 10bc -3b = 0 (6)
From equation (5), we can solve for b in terms of c: b = (51c - 8)/30. Substituting this into equation (6), it simplifies to:
(51c - 8)^2 - 90c(51c - 8) + 36c(51c - 8) = 0
After a wild circus of calculations, we find that c = -8/17. Substituting this back into b = (51c - 8)/30, we get b = -8/15.
So one possible function g(x) is g(x) = (-1/3)x^2 - (8/15)x - 8/17.
Now, if a = 1/3, we repeat the same process and find a different set of values for b and c. This leads to another possible function g(x).
Therefore, the sum of the coefficients in the quadratic function g(x) is either -1/3 - 8/15 - 8/17 OR 1/3 plus the corresponding coefficients from the other set.
To find the sum of the coefficients in the quadratic function g(x), we need to first express f(g(x)) in terms of g(x). Let's analyze the expression f(g(x)) = (5x^4)/9 + (17x^2)/3 + 21.
We know that f(g(x)) is equal to 5 times g(x) squared, minus 3 times g(x), plus 7. So, we can set up the following equation:
(5g(x)^2 - 3g(x) + 7) = (5x^4)/9 + (17x^2)/3 + 21
To solve this equation, we need to equate the corresponding terms on both sides:
5g(x)^2 = (5x^4)/9
-3g(x) = (17x^2)/3
7 = 21
Looking at the second equation, we can solve for g(x):
-3g(x) = (17x^2)/3
g(x) = -17x^2/9
Now, let's find the sum of the coefficients in g(x). The quadratic function g(x) can be written as g(x) = ax^2 + bx + c, where a, b, and c are the coefficients.
Comparing g(x) = -17x^2/9 with the general form, we can see that the sum of the coefficients is a + b + c = -17/9 + 0 + 0 = -17/9.
Therefore, the sum of the coefficients in the quadratic function g(x) is -17/9.