+8 1)Suppose f is a quadratic function defined by f(x) = ax^2 + bx + c for some numbers a,b,and c. If g(x) = x - 2 and f(g(x)) = 2x^2 - 5x + 19 for all values of x, what is the value of a + b + c?
2)If a and b are nonzero unequal real numbers and (a-b)/a = b/(a-b) what is the sum of all possible values for a/b?
+128408 1)Suppose f is a quadratic function defined by f(x) = ax^2 + bx + c for some numbers a,b,and c. If g(x) = x - 2 and f(g(x)) = 2x^2 - 5x + 19 for all values of x, what is the value of a + b + c?
f(g(x) must be
a(x - 2)^2 + b(x - 2) + c expand this
a(x^2 - 4x + 4) + bx - 2b + c
ax^2 - 4ax + 4a + bx - 2b + c
And we know that
ax^2 - 4ax + 4a + bx - 2b + c = 2x^2 - 5x + 19
ax^2 - ( 4a - b) x + (4a - 2b + c) = 2x^2 - 5x + 19
It's obvious that a = 2
And (4a - b) = 5
So
4(2) - b = 5
8 - b = 5
b = 3
And
4a - 2b + c = 19
4(2) -2(3) + c = 19
8 - 6 + c =19
2 + c =19
c = 17
So
a + b + c = 2 + 3 + 17 = 22
