We are given that the product of two numbers is 60, and their average is 8. We need to find the two numbers using quadratic equations.

1. Problem Description

We are given that the product of two numbers is 60, and their average is

8. We need to find the two numbers using quadratic equations.

2. Solution Steps

Let the two numbers be $x$ and $y$.

We are given that:

$xy = 60$ (1)

$\frac{x+y}{2} = 8$ (2)

From equation (2), we have:

$x+y = 16$

$y = 16 - x$ (3)

Substitute equation (3) into equation (1):

$x(16-x) = 60$

$16x - x^2 = 60$

$x^2 - 16x + 60 = 0$

Now we have a quadratic equation. We can solve it by factoring:

$(x-6)(x-10) = 0$

So, $x = 6$ or $x = 10$.

If $x = 6$, then $y = 16 - x = 16 - 6 = 10$.

If $x = 10$, then $y = 16 - x = 16 - 10 = 6$.

Thus, the two numbers are 6 and

1

0.

3. Final Answer

The two numbers are 6 and 10.