Find the value of `c+d` if `c-d=2,cd=63`.

Answer

To find the value of \( c + d \) given the equations \( c - d = 2 \) and \( cd = 63 \), we can follow these steps:

### Step 1: Start with the given equations
We have:
1. \( c - d = 2 \)
2. \( cd = 63 \)

### Step 2: Square the first equation
Square both sides of the first equation:
\[ (c - d)^2 = 2^2 \]
This gives us:
\[ c^2 - 2cd + d^2 = 4 \]

### Step 3: Substitute the value of \( cd \)
We know \( cd = 63 \). Substitute this into the equation:
\[ c^2 - 2(63) + d^2 = 4 \]
This simplifies to:
\[ c^2 - 126 + d^2 = 4 \]

### Step 4: Rearrange the equation
Rearranging gives:
\[ c^2 + d^2 = 4 + 126 \]
Thus:
\[ c^2 + d^2 = 130 \]

### Step 5: Use the identity for \( c + d \)
Now, we can find \( c + d \) using the identity:
\[ (c + d)^2 = c^2 + d^2 + 2cd \]
Substituting the known values:
\[ (c + d)^2 = 130 + 2(63) \]
Calculating \( 2(63) \):
\[ 2(63) = 126 \]
So we have:
\[ (c + d)^2 = 130 + 126 = 256 \]

### Step 6: Take the square root
Now, take the square root of both sides:
\[ c + d = \sqrt{256} \]
Since \( \sqrt{256} = 16 \):
\[ c + d = 16 \]

### Final Answer
Thus, the value of \( c + d \) is \( 16 \).

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