SOLUTION: the function f(x) = x^2 + 6x +20 + k(x^2 -3x -12) where k is real find the value of k and the value of p given that the minimum value of f(x) is p and that f(-2) =p the answers a

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: the function f(x) = x^2 + 6x +20 + k(x^2 -3x -12) where k is real find the value of k and the value of p given that the minimum value of f(x) is p and that f(-2) =p the answers a      Log On


   

Question 764499: the function f(x) = x^2 + 6x +20 + k(x^2 -3x -12) where k is real
find the value of k and the value of p given that the minimum value of f(x) is p and that f(-2) =p
the answers are 2/7 anf 80/7
i am happy if you just explain to me what to do, even briefly.
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
the function f(x) = x^2 + 6x +20 + k(x^2 -3x -12) where k is real
find the value of k and the value of p given that the minimum value of f(x) is p and that f(-2) =p
the answers are 2/7 and 80/7
i am happy if you just explain to me what to do, even briefly.

f(x) = x² + 6x + 20 + k(x² - 3x - 12)

f(x) = x² + 6x + 20 + kx² - 3kx - 12k

f(x) = x² + kx² + 6x - 3kx + 20 - 12k

f(x) = (1 + k)x² + (6 - 3k)x + 20 - 12k

We are told that f(-2) = p, so

f(-2) = (1 + k)(-2)² + (6 - 3k)(-2) + 20 - 12k = p

              (1 + k)(4) - 12  + 6k + 20 - 12k = p

                               4 + 4k + 8 - 6k = p

                                       12 - 2k = p

The minimum value of f(x) = ax² + bx + c is f%28-b%2F%282a%29%29

f(x) = (1 + k)x² + (6 - 3k)x + (20 - 12k)

-b%2F%282a%29 = %28-%286-3k%29%29%2F%282%281%2Bk%29%29

 f%28-b%2F%282a%29%29%22%22=%22%22f%28%28-%286-3k%29%29%2F%282%281%2Bk%29%29%29%22%22=%22%22%281%2Bk%29%28%28-%286-3k%29%29%2F%282%281%2Bk%29%29%29%5E2%22%22%2B%22%22%286+-+3k%29%28%28-%286-3k%29%29%2F%282%281%2Bk%29%29%29%22%22%2B%22%22%2820-12k%29%29%22%22=%22%22

                       %281%2Bk%29%28%28%286-3k%29%5E2%29%2F%284%281%2Bk%29%5E2%29%29%22%22-%22%22%28%286-3k%29%5E2%29%2F%282%281%2Bk%29%29%22%22%2B%22%22%2820-12k%29%29%22%22=%22%22

                       %28cross%281%2Bk%29%29%28%28%286-3k%29%5E2%29%2F%284%281%2Bk%29%5Ecross%282%29%29%29%22%22-%22%22%28%286-3k%29%5E2%29%2F%282%281%2Bk%29%29%22%22%2B%22%22%2820-12k%29%29%22%22=%22%22

                       %28%286-3k%29%5E2%29%2F%284%281%2Bk%29%29%22%22-%22%22%28%286-3k%29%5E2%29%2F%282%281%2Bk%29%29%22%22%2B%22%22%2820-12k%29%29


We are told this minimum value is equal to p, and we have above that

12 - 2k = p, so our equation is

%28%286-3k%29%5E2%29%2F%284%281%2Bk%29%29%22%22-%22%22%28%286-3k%29%5E2%29%2F%282%281%2Bk%29%29%22%22%2B%22%2220%22%22-%22%2212k%22%22=%22%2212%22%22-%22%222k

%28%286-3k%29%5E2%29%2F%284%281%2Bk%29%29%22%22-%22%22%28%286-3k%29%5E2%29%2F%282%281%2Bk%29%29%22%22=%22%22-8%22%22%2B%22%2210k

Multiply through by the LCD of 4(1 + k)

(6 - 3k)² - 2(6 - 3k)² = (-8 + 10k)4(1 + k)

            -(6 - 3k)² = 4(-8 + 10k)(1 + k)

     -(36 - 36k + 9k²) = 4(-8 + 2k + 10k²)

       -36 + 36k - 9k² = -32 + 8k + 40k²

       -49k² + 28k - 4 = 0

        49k² - 28k + 4 = 0

             (7k - 2)² = 0

                7k - 2 = 0
 
                    7k = 2

                     k = 2%2F7 

Then since 12 - 2k = p 

                     p = 12 - 2k 

                     p = 12 - 2%282%2F7%29

                     p = 84%2F7%22%22-%22%224%2F7

                     p = 80%2F7 

Edwin