The difference between the squares of two numbers is 120. The square of the smaller number is twice the greater number. Find the numbers.

Answer

To solve the problem, we will define the two numbers and set up equations based on the information given.

### Step 1: Define the Variables
Let the two numbers be \( x \) (the greater number) and \( y \) (the smaller number).

### Step 2: Set Up the Equations
From the problem statement, we have two pieces of information:
1. The difference between the squares of the two numbers is 120:
\[ x^2 - y^2 = 120 \]
2. The square of the smaller number is twice the greater number:
\[ y^2 = 2x \]

### Step 3: Substitute the Second Equation into the First
We can substitute \( y^2 \) from the second equation into the first equation:
\[ x^2 - (2x) = 120 \]

### Step 4: Rearrange the Equation
Rearranging gives us:
\[ x^2 - 2x - 120 = 0 \]

### Step 5: Factor the Quadratic Equation
Now we will factor the quadratic equation:
\[ (x - 12)(x + 10) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x - 12 = 0 \) → \( x = 12 \)
2. \( x + 10 = 0 \) → \( x = -10 \)

Since \( x \) represents the greater number, we will take \( x = 12 \).

### Step 7: Find \( y \)
Now, we can substitute \( x = 12 \) back into the second equation to find \( y \):
\[ y^2 = 2(12) = 24 \]
Taking the square root gives us:
\[ y = \sqrt{24} = 2\sqrt{6} \]

### Conclusion
The two numbers are \( x = 12 \) and \( y = 2\sqrt{6} \).