a, b, c and d are consecutive integers. If `d^(2) +b^(2) - c^(2) - a^(2) = 22`, then the numbers are

Options

A. 5,6,7,8
B. 4,5,6,7
C. 3,4,5,6
D. Can't say

Answer

To solve the problem, we need to find the consecutive integers \( a, b, c, \) and \( d \) such that the equation \( d^2 + b^2 - c^2 - a^2 = 22 \) holds true.

### Step-by-Step Solution:

1. **Define the Consecutive Integers:**
Let \( a = x \). Then, the consecutive integers can be defined as:
- \( b = x + 1 \)
- \( c = x + 2 \)
- \( d = x + 3 \)

2. **Substitute into the Equation:**
Substitute \( a, b, c, \) and \( d \) into the equation:
\[ d^2 + b^2 - c^2 - a^2 = 22 \]
becomes:
\[ (x + 3)^2 + (x + 1)^2 - (x + 2)^2 - x^2 = 22 \]

3. **Expand the Squares:**
Now, expand each term:
\[ (x + 3)^2 = x^2 + 6x + 9 \]
\[ (x + 1)^2 = x^2 + 2x + 1 \]
\[ (x + 2)^2 = x^2 + 4x + 4 \]
\[ x^2 = x^2 \]

4. **Combine All Terms:**
Substitute the expanded forms back into the equation:
\[ (x^2 + 6x + 9) + (x^2 + 2x + 1) - (x^2 + 4x + 4) - x^2 = 22 \]

5. **Simplify the Equation:**
Combine like terms:
\[ x^2 + 6x + 9 + x^2 + 2x + 1 - x^2 - 4x - 4 - x^2 = 22 \]
This simplifies to:
\[ (6x + 2x - 4x) + (9 + 1 - 4) = 22 \]
Which further simplifies to:
\[ 4x + 6 = 22 \]

6. **Solve for \( x \):**
Isolate \( x \):
\[ 4x = 22 - 6 \]
\[ 4x = 16 \]
\[ x = 4 \]

7. **Find the Values of \( a, b, c, d \):**
Now substitute \( x \) back to find \( a, b, c, d \):
- \( a = x = 4 \)
- \( b = x + 1 = 5 \)
- \( c = x + 2 = 6 \)
- \( d = x + 3 = 7 \)

Thus, the consecutive integers are \( 4, 5, 6, \) and \( 7 \).

### Final Answer:
The numbers are \( 4, 5, 6, 7 \).