Question

what the value of k
KXsq-10x-5<or=0 for all x?

Answer 1

k x^2 - 10 x - 5 = y

k x^2 - 10 x = y + 5

x^2 -(10/k)x = (1/k)(y+5)

x^2 -(10/k) x + 25/k^2 = y/k+ 5/k +(5/k)^2

(x-5/k)^2 = (1/k)(y+5 +25/k)
vertex at x = 5/k and y= -5-25/k
I want k negative so that the function is negative for large +or - x
I want the vertex at y = 0
-5 - 25/k = 0
25 / k = -5
k = -5
check my arithmetic !

Answer 2

-5 x^2 - 10 x - 5 = y

x^2 + 2 x + 1 = -y/5

(x+1)^2 = -(1/5)(y-0)
sure enough it works :)