If `(a+b)^(2)=9 and (a-b)^(2)=49`, what is the value of `a^(2)+b^(2)`?

Options

A. `17`
B. `20`
C. `29`
D. `58`

Answer

To solve the problem, we start with the given equations:

1. \((a + b)^2 = 9\)
2. \((a - b)^2 = 49\)

### Step 1: Expand both equations

We can expand these equations using the formula \((x + y)^2 = x^2 + y^2 + 2xy\) and \((x - y)^2 = x^2 + y^2 - 2xy\).

From the first equation:
\[ (a + b)^2 = a^2 + b^2 + 2ab = 9 \]

From the second equation:
\[ (a - b)^2 = a^2 + b^2 - 2ab = 49 \]

### Step 2: Set up the equations

Now we have two equations:
1. \(a^2 + b^2 + 2ab = 9\) (Equation 1)
2. \(a^2 + b^2 - 2ab = 49\) (Equation 2)

### Step 3: Add the two equations

Next, we will add Equation 1 and Equation 2:
\[ (a^2 + b^2 + 2ab) + (a^2 + b^2 - 2ab) = 9 + 49 \]
This simplifies to:
\[ 2(a^2 + b^2) = 58 \]

### Step 4: Solve for \(a^2 + b^2\)

Now, divide both sides of the equation by 2:
\[ a^2 + b^2 = \frac{58}{2} = 29 \]

### Final Answer

Thus, the value of \(a^2 + b^2\) is \(29\).

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