x^2 + kx + 3 and -x^2 + 4x + k
you want to find the max value of x^2 + kx + 3 and the min value of -x^2 + 4x + k and then set them equal to each other and then solve for k.
since these are in standard quadratic form, look for x = -b/2a in each.
x^2 + kx + 3 gets you:
a = 1
b = k
c = 3
x = -b/2a = -k/2
-x^2 + 4x + k gets you:
a = -1
b = 4
c = k
x = -b/2a gets you x = -4/-2 = 2
the min value of y for x^2 + kx + 3 will be at f(-b/2a) = f(-k/2).
the max value of y for -x^2 + 4x + k will be at f(-b/2a) = f(2).
f(-k/2) for x^2 + kx + 3 becomes (-k/2)^2 + k(-k/2) + 3.
simplify this equation to get:
f(-k/2) = -k^2/4 + 3
f(2) for -x^2 + 4x + k becomes -(2^2) + 4(2) + k.
simplify this equation to get:
f(2) = k + 4
f(-k/2) is the min value of x^2 + kx + 3
f(2) is the max value of -x^2 + 4x + k
set them equal to each other and you get:
f(-k/2) = f(2) which becomes:
-k^2/4 + 3 = k + 4
add k^2/4 to both sides of the equation and subtract 3 from both sides of the equation to get:
0 = k^2/4 + k + 4 - 3
simplify to get:
0 = k^2/4 + k + 1
multiply both sides of this equation by 4 to get:
0 = k^2 + 4k + 4
factor k^2 + 4k + 4 to get (k+2)*(k+2) = 0
solve for k to get k = -2.
the min point of x^2 + kx + 3 and the max point of -x^2 + 4x + k should be the same when k = -2.
replace k with -2 in x^2 + kx + 3 to get x^2 -2x + 3.
replace k with -2 in -x^2 + 4x + k to get -x^2 + 4x - 2.
min point of x^2 - 2x +3 is at (-b/2a,f(-b/2a))
-b/2a = 2/2 = 1
f(-b/2a) = f(1) = 1 - 2 + 3 = 2.
max point of -x^2 + 4x - 2 is at (-b/2a,f(-b/2a))
-b/2a = -4/-2 = 2
f(2) = -4 + 8 - 2 = 2.
min point of x^2 - 2x + 3 is at (1,2)
max point of -x^2 + 4x - 2 is at (2,2)
the y values are the same so the min value and max value are the same.
the graph of both equations is shown below:
