Question

When a two digit number is added to its reverse, the result is 143. The number is 3 less than the sum of the squares of its digits. Calculate this number.

Answer 1

let the number be  N =  a b    where  b is the units digit.  a = the tens digit.

the value of the number N = 10 * a + b
       number after reversing :  b a
the value of the number after reversing :  10 * b  + a

sum of the two numbers = 10 a + b + 10 b + a  = 11 (a + b)
   given,  11 ( a + b) = 143          =>  a + b = 13    --- (1)
   
   sum of squares of digits = a² + b²  = (a+b)² - 2 a b
   given,      N = 10 a + b  = (a² + b²) - 3
          =>   10 a + b =  (13² - 2 a b) - 3 =  166 - 2 a b
    Hence,  10 a + b + 2 a b =  166      --- (2)

  do  (2) - 10 * (1) :  - 9 b + 2 a b = 36
            so       b (2a -9) = 36    --- (4) .

     substitute value of b from (1) in (4) :
             (13 - a) ( 2a - 9) = 36
     =>   -2 a²  + 35 a - 117 = 36
       =>    2 a² - 35 a + 153 = 0
          discr:  35² - 8 * 153 = 1
             a = (35 +- 1)/ 4  =  9  or  8.5
  we take the integer value only as a and b are integers less than 10.
        so a = 9 and   b = 4

so the number is  94.

Answer 2

let the tens digit of the no. be 'x'. and ones digit be 'y'
the no. will be 10x+y
according to the question,,10x+y + 10y+x=143
                                       11x+11y=143
by solving we get             x+y=13
                                       y=x-13   ..............eq 1
 
  according to the question,,10x+y+3=x²+y² ............................eq 2
  substituting eq 1 in eq 2
                                        10x+13-x+3=x²+(13-x)²
 by solving we get              2x²-35x+153=0
 
Solving the quadratic equation
a=2          b=-35             c=153
determinant=b²-4ac
                 =1225-1224
                 =1
          √D   =1

x=-b+√D÷2a                               or        x=-b-√D÷2a
x=9                                           or        x=34/4(ruled out since digits cannot 
                                                                        be fractions)
hence   x=9
            y=13-9
             =4
the number is 94.

hope this helps u...