Form the quadratic if the roots are 5/3 and 3/5

We'll use Viete's relations and we'll form the quadratic
when we know the sum and the product of the roots:

x1 + x2
= 5/3 + 3/5

x1 + x2 = (25 +
9)/15

x1 + x2 =
34/15

S = x1 + x2 =
34/15

x1*x2 =
(5/3)*(3/5)

x1*x2 =
1

P =
1

We'll form the quadratic, knowing the
values of the sum and the product:

x^2 - Sx + P =
0

x^2 - 34x/15 + 1 = 0

We'll
multiply by 15 and we'll get the final form of the quadratic whose sum is 34/15 and
product is 1.

15x^2 - 34x + 15 =
0

We also could use the fact
that a quadratic equation could be written as a product of linear factors, when the
solutions are given.

ax^2 + bx + c = (x -
5/3)(x -3/5)

We'll remove the
brackets;

ax^2 + bx + c = x^2 - 3x/5 - 5x/3 +
1

We'll combine like terms and
we'll
get:

ax^2
+ bx + c = x^2 - 34x/15 +
1