Determine the coefficients of the quadratic equation with roots 1/3 and 3.

The quadratic equation with roots 1/3 and 3 has coefficients of 9 and -10.

To find the quadratic equation with roots 1/3 and 3, we start by using the fact that the equation can be written in the form (x - r1)(x - r2) = 0, where r1 and r2 are the roots. Substituting in the given roots, we get:

(x - 1/3)(x - 3) = 0

Expanding the brackets, we get:

x^2 - (1/3 + 3)x + 1/3 * 3 = 0

Simplifying, we get:

x^2 - (10/3)x + 1 = 0

Therefore, the coefficients of the quadratic equation are 1, -10/3, and 1. However, we can simplify this by multiplying all the coefficients by 3 to get rid of the fraction:

3x^2 - 10x + 3 = 0

So the coefficients of the quadratic equation with roots 1/3 and 3 are 3, -10, and 3, or equivalently, 9, -30, and 9. We can simplify this further by dividing all the coefficients by 3:

x^2 - (10/3)x + 1 = 0

Therefore, the coefficients of the quadratic equation with roots 1/3 and 3 are 9 and -10.