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 and 80/7

i am happy if you just explain to me what to do, even briefly.
<pre>

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(-b/(2a))}}}

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

{{{-b/(2a)}}} = {{{(-(6-3k))/(2(1+k))}}}

 {{{f(-b/(2a))}}}{{{""=""}}}{{{f((-(6-3k))/(2(1+k)))}}}{{{""=""}}}{{{(1+k)((-(6-3k))/(2(1+k)))^2}}}{{{""+""}}}{{{(6 - 3k)((-(6-3k))/(2(1+k)))}}}{{{""+""}}}{{{(20-12k))}}}{{{""=""}}}

                       {{{(1+k)(((6-3k)^2)/(4(1+k)^2))}}}{{{""-""}}}{{{((6-3k)^2)/(2(1+k))}}}{{{""+""}}}{{{(20-12k))}}}{{{""=""}}}

                       {{{(cross(1+k))(((6-3k)^2)/(4(1+k)^cross(2)))}}}{{{""-""}}}{{{((6-3k)^2)/(2(1+k))}}}{{{""+""}}}{{{(20-12k))}}}{{{""=""}}}

                       {{{((6-3k)^2)/(4(1+k))}}}{{{""-""}}}{{{((6-3k)^2)/(2(1+k))}}}{{{""+""}}}{{{(20-12k))}}}


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

12 - 2k = p, so our equation is

{{{((6-3k)^2)/(4(1+k))}}}{{{""-""}}}{{{((6-3k)^2)/(2(1+k))}}}{{{""+""}}}{{{20}}}{{{""-""}}}{{{12k}}}{{{""=""}}}{{{12}}}{{{""-""}}}{{{2k}}}

{{{((6-3k)^2)/(4(1+k))}}}{{{""-""}}}{{{((6-3k)^2)/(2(1+k))}}}{{{""=""}}}{{{-8}}}{{{""+""}}}{{{10k}}}

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 = {{{2/7}}} 

Then since 12 - 2k = p 

                     p = 12 - 2k 

                     p = 12 - 2{{{(2/7)}}}

                     p = {{{84/7}}}{{{""-""}}}{{{4/7}}}

                     p = {{{80/7}}} 

Edwin</pre>