We need to determine the quadratic equation whose roots are 3 and 5.

There are two ways to find the equation.

We will use the factors method to determine the function.

We find the factors of the quadratic function.

Let f(x) be the function where 3 and 5 are the roots.

==> Then, the factors are (x-3) and (x-5)

==> f(x) = (x-3)(x-5)

We will open the brackets.

==> f(x) = x^2 -3x -5x + 15

==>** f(x) = x^2 - 8x + 15**

The roots of the quadratic equation are x= 3 and x = 5

So we can write: (x - 3)(x - 5) = 0

=> x^2 - 3x - 5x + 15 = 0

=> x^2 - 8x + 15 = 0

**The required quadratic equation is x^2 - 8x + 15 = 0**

The quadratic equation whose roots are x= x1 and x= x2 is (x-x1)(x-x2) = 0. Or x^2 -(x1+x2)x+x1x2 = 0.

So the quadratic equation whose roots are x= 3 and x = 5 is obtained by the product (x-3)(x-5) = 0.

=> x(x-5) -3(x-5) = 0

=> x^2-5x-3x + 3*5 = 0.

=> x^2-(3+5)x+ 3*5 = 0.

=> x^2-8x +15 = 0.

**Therefore the quadratic equaltion is x^2-8x+15 = 0 **