The product of two numbers is 120. The sum of their squares is 289. The difference of these two numbers is

Options

A. 9
B. 7
C. 8
D. 6

Answer

To solve the problem, we need to find two numbers \( x \) and \( y \) such that:

1. Their product is given by:
\[ xy = 120 \]

2. The sum of their squares is given by:
\[ x^2 + y^2 = 289 \]

We want to find the difference between the two numbers, \( |x - y| \).

### Step 1: Use the identity for the sum of squares
We can use the identity that relates the sum of squares to the product and the sum of the numbers:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Substituting the known values into this equation:
\[ 289 = (x + y)^2 - 2(120) \]
This simplifies to:
\[ 289 = (x + y)^2 - 240 \]

### Step 2: Solve for \( (x + y)^2 \)
Rearranging the equation gives:
\[ (x + y)^2 = 289 + 240 \]
\[ (x + y)^2 = 529 \]

### Step 3: Take the square root
Taking the square root of both sides, we find:
\[ x + y = \sqrt{529} = 23 \]

### Step 4: Set up a system of equations
Now we have two equations:
1. \( x + y = 23 \)
2. \( xy = 120 \)

### Step 5: Use substitution to find \( x \) and \( y \)
We can express \( y \) in terms of \( x \):
\[ y = 23 - x \]
Substituting this into the product equation:
\[ x(23 - x) = 120 \]
Expanding this gives:
\[ 23x - x^2 = 120 \]
Rearranging leads to:
\[ x^2 - 23x + 120 = 0 \]

### Step 6: Solve the quadratic equation
Now we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -23, c = 120 \):
\[ x = \frac{23 \pm \sqrt{(-23)^2 - 4 \cdot 1 \cdot 120}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (-23)^2 - 480 = 529 - 480 = 49 \]
So we have:
\[ x = \frac{23 \pm \sqrt{49}}{2} \]
\[ x = \frac{23 \pm 7}{2} \]

### Step 7: Find the values of \( x \)
Calculating the two possible values:
1. \( x = \frac{30}{2} = 15 \)
2. \( x = \frac{16}{2} = 8 \)

Thus, the two numbers are \( x = 15 \) and \( y = 8 \).

### Step 8: Calculate the difference
Now, we can find the difference:
\[ |x - y| = |15 - 8| = 7 \]

### Final Answer
The difference of the two numbers is:
\[ \boxed{7} \]