Suppose there are 100 strings, each of the string, of course, has 2 ends. then you randomly choose 2 of the ends and tie them together. each end will be tied only once and the process repeats until there are no free ends left. (i. e. it will lead to 100 randomly chosen pairs of tied ends.) let l be the number of resulting loops. what is e(l)?

x^2 + 81

we can factor this using difference of square binomials.

rewrite the equation:

x^2 - (-81)

factor:

(a^2 - b^2) = (√a^2 + √b^2)(√a^2 - √b^2)

(x^2 - (-81) = (x + √-81)(x - √-81)

we can remove the - from under the radical by multiplying it by i.

(x^2 - (-81) = (x + i√81)(x - i√81)

(x^2 - (-81) = (x + 9i)(x - 9i)

we can check our answer by using foil:

x^2 - 9ix + 9ix - 81i^2

x^2 - 81^2

i^2 = -1

x^2 + 81

c and d is your answers.

stay frosty!