Question 592664
>>...two children whose ages differ by 5 years...<<
<pre>
Let the younger child's age = x
Then the older child's age = x+5
</pre>
>>...The sum of the squares of their ages is 97...<<
<pre>
           x² + (x+5)² = 97

       x² + (x+5)(x+5) = 97

x² + x² + 5x + 5x + 25 = 97
    
        2x² + 10x + 25 = 97

        2x² + 10x - 72 = 0

Divide through by 2

          x² + 5x - 36 = 0

Factor

        (x + 9)(x - 4) = 0

Use the zero-factor principle:

   x + 9 = 0;       x - 4 = 0  
       x = -9;          x = 4

Ignore the negative answer. 

The younger child is 4.
The older child is 5 years older or 9.
</pre>
>>...The square of the mother’s age can be found by writing 
the square of the children’s ages one after the other as a four-digit number...<<
<pre>
The square of the younger child's age = 4² = 16

The square of the older child's age = 9² = 81

Four-digit number = 1681 so,

The square of the mother's age is 1681.

The mother's age =  {{{sqrt(1681)}}} = 41

Edwin</pre>