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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?

 Mar 21, 2020
 #1
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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

 

 

cool cool cool

 Mar 21, 2020
 #2
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Can Someone post the solution to #2?

 Mar 28, 2020

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