If the sum of two numbers added to the sum of their squares is 42 and the product of these numbers is 15, then the numbers are :

Options

A. 15, 1
B. `15/6, 6`
C. `2(1)/2, 6`
D. 5, 3

Answer

To solve the problem step by step, we can denote the two numbers as \( x \) and \( y \). We are given two conditions:

1. The sum of the two numbers added to the sum of their squares is 42:
\[ x + y + x^2 + y^2 = 42 \]

2. The product of the two numbers is 15:
\[ xy = 15 \]

### Step 1: Express the sum of squares in terms of the sum and product
We know that:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Let \( s = x + y \) (the sum of the numbers) and \( p = xy \) (the product of the numbers). From the second condition, we have \( p = 15 \).

Substituting this into the equation for the sum of squares gives:
\[ x^2 + y^2 = s^2 - 2p = s^2 - 30 \]

### Step 2: Substitute into the first equation
Now we can substitute \( x^2 + y^2 \) into the first equation:
\[ s + (s^2 - 30) = 42 \]
This simplifies to:
\[ s + s^2 - 30 = 42 \]
\[ s^2 + s - 72 = 0 \]

### Step 3: Solve the quadratic equation
Now we need to solve the quadratic equation \( s^2 + s - 72 = 0 \). We can use the quadratic formula:
\[ s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = 1, c = -72 \).

Calculating the discriminant:
\[ b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-72) = 1 + 288 = 289 \]
Now substituting into the formula:
\[ s = \frac{-1 \pm \sqrt{289}}{2 \cdot 1} = \frac{-1 \pm 17}{2} \]
Calculating the two possible values for \( s \):
1. \( s = \frac{16}{2} = 8 \)
2. \( s = \frac{-18}{2} = -9 \) (not possible since \( x \) and \( y \) are positive)

Thus, \( s = 8 \).

### Step 4: Find the numbers using the sum and product
Now we have:
- \( s = x + y = 8 \)
- \( p = xy = 15 \)

We can use these to form a new quadratic equation:
\[ t^2 - st + p = 0 \implies t^2 - 8t + 15 = 0 \]

### Step 5: Solve for \( t \)
Using the quadratic formula again:
\[ t = \frac{8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot 15}}{2 \cdot 1} = \frac{8 \pm \sqrt{64 - 60}}{2} = \frac{8 \pm \sqrt{4}}{2} = \frac{8 \pm 2}{2} \]
Calculating the two possible values for \( t \):
1. \( t = \frac{10}{2} = 5 \)
2. \( t = \frac{6}{2} = 3 \)

Thus, the two numbers are \( 5 \) and \( 3 \).

### Final Answer:
The numbers are \( 5 \) and \( 3 \).