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

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