Question

The sum of the squares of three consecutive integers is 194. What are the integers?

Answer 1

Consecutive integers are 1 apart
they are
x,x+1,x+2
the sum of their squares is 194

x²+(x+1)²+(x+2)²=194
expand
x²+x²+2x+1+x²+4x+4=194
x²+x²+x²+2x+4x+1+4=194
3x²+6x+5=194
minus 5 from both sides
3x²+6x=189
3(x²+2x)=189
divide both sides by 3
x²+2x=63
minus 63 from bot sides
x²+2x-63=0
what 2 numbers multiply to get -63 and add to get 2
-7 and 9
(x-7)(x+9)=0
set each to zero

x-7=0
x=7

x+9=0
x=-9

for x=7
x+1=8
x+2=9

for x=-9
x+1=-8
x+2=-7

the integers are 7,8,9 or -9,-8,-7