Hi  
Katherine is younger than Alexandra. 
Their ages are consecutive integers. 
Let Katherine age be  x and Alexandra's (x+1)
the sum of the square of Katherine's age and 5 times Alexandra's
age is 109.
Find Katherine's age:
 x^2 + 5(x+1) = 109
 x^2 + 5x - 104 = 0
 (x + 13 )(x -  8) = 0   |Note: x = -13, not appropriate for age
           x = 8, Katherine's age.
checking our work:
 8^2 + 5*9 = 64 + 45 = 109  checks.
  
Wish You the Best in your Studies.