Two friend are $$ x $$ and $$ y $$ years old. The sum of their age is 36 . In five yeas time the Product of their age will be 528 . Fund $$ x $$ and

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1. We are given that the sum of their ages is  
   $$
   x + y = 36.
   $$
2. In five years, their ages will be $$ x + 5 $$ and $$ y + 5 $$. The product of their ages in five years is given by  
   $$
   (x+5)(y+5) = 528.
   $$
3. Expanding the product, we have  
   $$
   (x+5)(y+5) = xy + 5x + 5y + 25.
   $$
4. From step 1, since $$ x + y = 36 $$, we substitute to obtain  
   $$
   xy + 5(36) + 25 = xy + 180 + 25 = xy + 205.
   $$
   Setting the expression equal to 528, we write  
   $$
   xy + 205 = 528.
   $$
5. Solving for $$ xy $$, we subtract 205 from both sides:  
   $$
   xy = 528 - 205 = 323.
   $$
6. We now have a system with the sum and product of $$ x $$ and $$ y $$:
   $$
   x + y = 36 \quad \text{and} \quad xy = 323.
   $$
   The corresponding quadratic equation whose roots are $$ x $$ and $$ y $$ is  
   $$
   t^2 - (x+y)t + xy = 0,
   $$
   or
   $$
   t^2 - 36t + 323 = 0.
   $$
7. We solve the quadratic equation using the quadratic formula  
   $$
   t = \frac{36 \pm \sqrt{36^2 - 4 \cdot 323}}{2}.
   $$
8. First, compute the discriminant:  
   $$
   36^2 - 4 \cdot 323 = 1296 - 1292 = 4.
   $$
9. Since the discriminant is positive, the roots are  
   $$
   t = \frac{36 \pm \sqrt{4}}{2} = \frac{36 \pm 2}{2}.
   $$
10. Calculating both solutions:
    - When using the positive square root:  
      $$
      t = \frac{36 + 2}{2} = \frac{38}{2} = 19.
      $$
    - When using the negative square root:  
      $$
      t = \frac{36 - 2}{2} = \frac{34}{2} = 17.
      $$
11. Thus, the two friends are $$ 17 $$ and $$ 19 $$ years old.

So, the solution for $$ x $$ is either $$ 17 $$ or $$ 19 $$, and accordingly, $$ y $$ is $$ 19 $$ or $$ 17 $$.
The two friends are 17 and 19 years old.

Two friend are \( x \) and \( y \) years old. The sum of their age is 36 . In five yeas time the Product of their age will be 528 . Fund \( x \) and

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