Let two numbers have arithmetic mean $$9$$ and geometric mean $$4$$. Then these numbers are the roots of the quadratic equation

Let the two numbers be $$ a $$ and $$ b $$. The arithmetic mean of these two numbers is given as 9:

$$ \frac{a + b}{2} = 9 $$

Multiplying both sides by 2 gives the sum of the roots:

$$ a + b = 18 $$

The geometric mean of the two numbers is given as 4:

$$ \sqrt{ab} = 4 $$

Squaring both sides gives the product of the roots:

$$ ab = 16 $$

Any quadratic equation with roots $$ a $$ and $$ b $$ can be written using the formula:

$$ x^2 - (a + b)x + ab = 0 $$

Substitute the values of the sum and the product into the formula:

$$ x^2 - 18x + 16 = 0 $$

Final Answer:

The required quadratic equation is $$ x^2 - 18x + 16 = 0 $$.