Find a two-digit number such that the sum of the digits is 12 and the sum of their squares is 74. 

Given, the Sum of the digits is 12. 

Let one digit be x. Another digit is 12 - x.

As the Sum of their squares = 74 

$\Rightarrow {\mathrm{x}}^2+(12-\mathrm{x}{)}^2=74$         

$\Rightarrow {\mathrm{x}}^2+144{ +\mathrm{x}}^2-24\mathrm{x}=74$ 

$\Rightarrow {2\mathrm{x}}^2-24\mathrm{x}+144-74=0$  

$\Rightarrow {\mathrm{x}}^2-12\mathrm{x}+35=0$ 

$\Rightarrow {\mathrm{x}}^2-7\mathrm{x}-5\mathrm{x}+35+0$

=> x(x - 7) - 5(x - 7) = 0

=> (x - 7)(x - 5) = 0

=> x = 7, 5

Hence, The number is 57 or 75.