If a + b = 9 and ab = 10, find the value of a^2+b^2.

Answer

To find the value of \( a^2 + b^2 \) given that \( a + b = 9 \) and \( ab = 10 \), we can follow these steps:

### Step 1: Use the identity for \( a^2 + b^2 \)
We know that:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
This identity allows us to express \( a^2 + b^2 \) in terms of \( a + b \) and \( ab \).

### Step 2: Substitute the known values
From the problem, we have:
- \( a + b = 9 \)
- \( ab = 10 \)

Now we can substitute these values into the identity:
\[ a^2 + b^2 = (9)^2 - 2(10) \]

### Step 3: Calculate \( (a + b)^2 \)
Calculating \( (9)^2 \):
\[ (9)^2 = 81 \]

### Step 4: Calculate \( 2ab \)
Calculating \( 2(10) \):
\[ 2(10) = 20 \]

### Step 5: Substitute back into the equation
Now substitute these results back into the equation:
\[ a^2 + b^2 = 81 - 20 \]

### Step 6: Simplify the expression
Now, simplify the right side:
\[ a^2 + b^2 = 61 \]

### Final Answer
Thus, the value of \( a^2 + b^2 \) is \( 61 \).

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