Two numbers differ by 8 and their product is 153. The number are:

Options

A. `9,-17`
B. `-17,9`
C. `17,9`
D. `11,19`

Answer

To solve the problem where two numbers differ by 8 and their product is 153, we can follow these steps:

### Step 1: Set up the equations
Let the two numbers be \( x \) and \( y \). According to the problem:
1. The difference between the two numbers is given by:
\[ x - y = 8 \quad \text{(Equation 1)} \]
2. The product of the two numbers is given by:
\[ xy = 153 \quad \text{(Equation 2)} \]

### Step 2: Express one variable in terms of the other
From Equation 1, we can express \( x \) in terms of \( y \):
\[ x = y + 8 \]

### Step 3: Substitute into the second equation
Now, substitute \( x \) in Equation 2:
\[ (y + 8)y = 153 \]
Expanding this gives:
\[ y^2 + 8y = 153 \]

### Step 4: Rearrange the equation
Rearranging the equation to set it to zero:
\[ y^2 + 8y - 153 = 0 \]

### Step 5: Factor the quadratic equation
Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to \(-153\) and add to \(8\). The factors of \(-153\) that satisfy this are \(17\) and \(-9\):
\[ (y + 17)(y - 9) = 0 \]

### Step 6: Solve for \( y \)
Setting each factor to zero gives us:
1. \( y + 17 = 0 \) → \( y = -17 \)
2. \( y - 9 = 0 \) → \( y = 9 \)

### Step 7: Find corresponding \( x \) values
Now we can find the corresponding \( x \) values for both \( y \) values:
1. If \( y = 9 \):
\[ x = 9 + 8 = 17 \]
2. If \( y = -17 \):
\[ x = -17 + 8 = -9 \]

### Conclusion
The two pairs of numbers that satisfy the conditions of the problem are:
1. \( (17, 9) \)
2. \( (-9, -17) \)