What will come in place of question mark(?) in the following questions?
`(45)^2 + (?)^2 =(47)^2 +257`

Options

A. 23
B. 21
C. 441
D. 529

Answer

To solve the equation \( (45)^2 + (?)^2 = (47)^2 + 257 \), we will follow these steps:

### Step 1: Calculate \( (45)^2 \) and \( (47)^2 \)
First, we need to find the squares of 45 and 47.

\[ (45)^2 = 2025 \]
\[ (47)^2 = 2209 \]

### Step 2: Substitute the values into the equation
Now, we substitute these values into the original equation:

\[ 2025 + (?)^2 = 2209 + 257 \]

### Step 3: Calculate \( 2209 + 257 \)
Next, we add \( 2209 \) and \( 257 \):

\[ 2209 + 257 = 2466 \]

### Step 4: Set up the equation
Now, we can rewrite our equation:

\[ 2025 + (?)^2 = 2466 \]

### Step 5: Isolate \( (?)^2 \)
To find \( (?)^2 \), we will isolate it by subtracting \( 2025 \) from both sides:

\[ (?)^2 = 2466 - 2025 \]

### Step 6: Calculate \( 2466 - 2025 \)
Now, we perform the subtraction:

\[ 2466 - 2025 = 441 \]

### Step 7: Find the value of \( ? \)
Now we have:

\[ (?)^2 = 441 \]

To find \( ? \), we take the square root of both sides:

\[ ? = \sqrt{441} \]

### Step 8: Calculate \( \sqrt{441} \)
Calculating the square root gives us:

\[ ? = 21 \]

Thus, the value that comes in place of the question mark is \( 21 \).

### Final Answer:
\[ ? = 21 \]

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