If `a-b= 4 and a+b= 6`, find:
`a^(2) + b^(2)`

Answer

To solve the problem where we need to find \( a^2 + b^2 \) given the equations \( a - b = 4 \) and \( a + b = 6 \), we can follow these steps:

### Step-by-Step Solution:

1. **Write down the equations:**
We have:
\[ a - b = 4 \quad \text{(1)} \]
\[ a + b = 6 \quad \text{(2)} \]

2. **Add the two equations:**
By adding equations (1) and (2), we can eliminate \( b \):
\[ (a - b) + (a + b) = 4 + 6 \]
This simplifies to:
\[ 2a = 10 \]

3. **Solve for \( a \):**
Divide both sides by 2:
\[ a = \frac{10}{2} = 5 \]

4. **Substitute \( a \) back into one of the original equations to find \( b \):**
Using equation (2):
\[ a + b = 6 \]
Substitute \( a = 5 \):
\[ 5 + b = 6 \]
Solving for \( b \):
\[ b = 6 - 5 = 1 \]

5. **Now calculate \( a^2 + b^2 \):**
We have \( a = 5 \) and \( b = 1 \):
\[ a^2 + b^2 = 5^2 + 1^2 \]
Calculate:
\[ = 25 + 1 = 26 \]

### Final Answer:
Thus, the value of \( a^2 + b^2 \) is \( 26 \).

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