At present Asha's age (in years) is 2 more than the square of her daughter Nisha's age. When Nisha grows to her mother's present age, Asha's age would be one year less than 10 times the present age of Nisha. Find the present age of both Asha and Nisha.

Problem : At present Asha's age (in years) is 2 more than the square of her daughter Nisha's age. When Nisha grows to her mother's present age, Asha's age would be one year less than 10 times the present age of Nisha. Find the present age of both Asha and Nisha.

Doubt by Muskan

Solution : 

Let the present age of Nisha (Daughter) = x years
and the present age of Asha (Mother) = (2 + x²) years

Difference between the ages of Mother and Daughter
= (2+x²) - x

When Age of Nisha becomes equal to the present age of her mother (Asha) i.e (2 + x²) then the present age of her mother (Asha) would be = [(2 + x²) + (2 + x²) - x]

Future age of Asha's (Mother) = 10 (Present age of Nisha) - 1
[(2 + x²) + (2 + x²) - x] = 10x - 1
4 + 2x² - x = 10x + 1  
2x² - x - 10x + 4 + 1 = 0
2x² - 11x + 5 = 0
2x² - (10 + 1)x + 5 = 0
2x² - 10x - 1x + 5 = 0
2x (x - 5) -1 (x - 5) = 0
(2x - 1) (x - 5) = 0

2x - 1 = 0 OR x - 5 = 0
2x = 1 OR X = 5 
x = 1/2 OR x = 5 

When x = 1/2 = 0.5

The present age of Nisha (Daughter) = 0.5  years
and the present age of Asha (Mother)
= (2 + (0.5)²) years
= 2.25 years
which is not possible. 

When x = 5

The present age of Nisha (Daughter) = 5 years
and the present age of Asha (Mother)
= (2 + x²) years 

= 2 + (5)²
= 27 years