1. the purpose of this exercise is to provide practice using the lingo or excel solvers. find the values of x and y that minimize the function

min x 2 - 4x + y 2 + 8y + 20

do not assume nonnegativity of the x and y variables. recall that by default lingo assumes nonnegative variables. in order to allow the variables to take on negative values you can add

@free(x); @free(y);

alternatively, if you want lingo to allow for negative values by default, in the lingo menu select options and then click general solver, and then uncheck the variables assumed nonnegative tab.

Find the minmum value of x^2-4x+y^2+8y+20=?
we don't know what it's equal to, so we set it equal to ?, if you know that ? is, then input it
what we can do is try to convert it to a conic section
complete the squares

(x^2-4x)+(y^2+8y)+20=?
take 1/2 of linear coefient and square it and add negative and positive insde (linear is 1st degree)

(x^2-4x+4-4)+(y^2+8x+16-16)+20=?
factor perfect squares
((x-2)^2-4)+((y+4)^2-16)+20=?
(x-2)^2-4+(y+4)^2-16+20=?
(x-2)^2+(y+4)^2=?

we see this is the equation of a circle
in form
(x-h)^2+(y-k)^2=r^2
center is (h,k)
radius is r
so the lowest point is r  units down from (h,k), or the point (h,k-r)

we know that te equation is
(x-2)^2+(y-(-4))^2=?
?=r^2
√?=r
center is (2,-4)
therefor
the minimum value, where the equation is equal to ?, is (2,-4-√?)
good luck, and may the force be with you