# If natural numbers #a, b, c, d# satisfy #a^2+b^2 = 41# and #c^2+d^2=25# then what monic quadratic in #x# has zeros #(a+b)# and #(c+d)# ?

##### 1 Answer

#### Answer:

#### Explanation:

I will assume that "natural number" includes

We may suppose that

Hence:

#a^2 <= 41/2" "# so#" "a = 0, 1, 2, 3# or#4" "# and#" "a^2 = 0, 1, 4, 9# or#16# .

#b^2 = 41 - a^2 = color(red)(cancel(color(black)(41))), color(red)(cancel(color(black)(40))), color(red)(cancel(color(black)(37))), color(red)(cancel(color(black)(32)))# or#25# .

#c^2 <= 25/2" "# so#" "c = 0, 1, 2# or#3" "# and#" "c^2 = 0, 1, 4# or#9# .

#d^2 = 25-c^2 = 25, color(red)(cancel(color(black)(24))), color(red)(cancel(color(black)(21)))# or#16# .

So

So

So the polynomial we are looking for is one of:

#(x-9)(x-5) = x^2-14x+45#

#(x-9)(x-7) = x^2-16x+63#