If the number p is 5 more than q and the sum of the squares of p and q is 55, then the product of p and q is

Options

A. 10
B. `-10`
C. 15
D. `-15`

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Set up the equations
We know from the problem statement that:
1. \( p = q + 5 \) (since p is 5 more than q)
2. \( p^2 + q^2 = 55 \) (the sum of the squares of p and q)

### Step 2: Substitute p in the second equation
We can substitute the expression for \( p \) from the first equation into the second equation:
\[ (q + 5)^2 + q^2 = 55 \]

### Step 3: Expand the equation
Now, we will expand \( (q + 5)^2 \):
\[ (q^2 + 10q + 25) + q^2 = 55 \]
This simplifies to:
\[ 2q^2 + 10q + 25 = 55 \]

### Step 4: Rearrange the equation
Next, we will rearrange the equation to set it to zero:
\[ 2q^2 + 10q + 25 - 55 = 0 \]
This simplifies to:
\[ 2q^2 + 10q - 30 = 0 \]

### Step 5: Divide the entire equation by 2
To simplify the equation, we can divide everything by 2:
\[ q^2 + 5q - 15 = 0 \]

### Step 6: Factor the quadratic equation
Now, we need to factor the quadratic equation:
\[ (q + 8)(q - 3) = 0 \]
This gives us two possible solutions for \( q \):
1. \( q = -8 \)
2. \( q = 3 \)

### Step 7: Find the corresponding values of p
Using \( p = q + 5 \):
1. If \( q = -8 \), then \( p = -8 + 5 = -3 \).
2. If \( q = 3 \), then \( p = 3 + 5 = 8 \).

### Step 8: Calculate the product of p and q
Now we can calculate the product \( p \times q \):
1. For \( q = -8 \) and \( p = -3 \):
\[ p \times q = -3 \times -8 = 24 \]
2. For \( q = 3 \) and \( p = 8 \):
\[ p \times q = 8 \times 3 = 24 \]

### Conclusion
In both cases, the product \( p \times q = 24 \).

### Final Answer
The product of \( p \) and \( q \) is **24**.