SAT Math Problem (24)

Question

If m and n are integers such that m²+n²=40 and m< 0 < n, what is the value of (m+n)²?

Answer 1

 

   m² + n² = 40

   

m is q negative integer and n is positive integer

 

possible values for m ... -1, -2. -2. -3. -4, etc

possible values for n ... 1, 2, 3, 4, 5, 6, etc.

 

if n is 6 and m is -2.  we have

 

   (-2)² + 6² = 4 + 36 = 40

 

Thus (m + n)² = (-2+6)² = 4² = 16

 

Answer 2

m² + n² = 40

 

m² = 40 - n²

m = -√(40 - n²), since m < 0

 

Since  m < 0, then -√(40 - n²) < 0

 

Square both sides, 0 < 40 - n²

                           n² < 40

                       0 < n < 2√(10) ≈ 6.3

 

n ∈ {1, 2, 3, 4, 5, 6}

 

Then the set of corresponding values for m are

m = {-√(39), -6, -√(31), -2√(6), -√(15), -2}

 

There are two possible solutions in integers,

(m, n) = (-6,2) or (-2, 6)

 

In either case (m + n) ² = (±4)² = 16