Question

please answer question with full explanation.
Q1 Sum of squares of three consecutive numbers is 869. Find the middle number

Answer 1

let the first number is x
then next will be x+1
and x+2
${x}^{2} + ( {x + 1)}^{2} + ( {x + 2)}^{2} = 869 \\ {x}^{2} + {x}^{2} + 2x + 1 + {x}^{2} + 4x + 4 = 869 \\ 3 {x}^{2} + 6x + 5 - 869 = 0 \\ 3 {x}^{2} + 6x - 864 = 0 \\ {x}^{2} + 2x - 288 = 0 \\ {x}^{2} + 18x - 16x - 288 = 0 \\ x(x + 18) - 16(x + 18) = 0 \\ (x - 16)(x + 18) = 0 \\ x - 16 = 0 \\ x = 16 \\ x + 18 = 0 \\ x = - 18$
you can discard negative number.
now numbers are 16,17,18.
middle number is 17.
verification:
$= ( {16)}^{2} + ( {17)}^{2} + ( {18)}^{2} \\ = 256 + 289 + 324 \\ = 869$