If `(p-q)^(2)=25 and pq=14`, what is the values of `(p+q)^(2)`?

Options

A. `25`
B. `36`
C. `53`
D. `81`

Answer

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

1. \((p - q)^2 = 25\)
2. \(pq = 14\)

We need to find the value of \((p + q)^2\).

### Step 1: Use the identity for the square of a sum and difference
We know that:
\[ (p + q)^2 - (p - q)^2 = 4pq \]
This identity relates the squares of the sum and difference of two numbers to their product.

### Step 2: Substitute the known values
From the problem, we have:
- \((p - q)^2 = 25\)
- \(pq = 14\)

Substituting these values into the identity:
\[ (p + q)^2 - 25 = 4 \times 14 \]

### Step 3: Calculate \(4pq\)
Now, calculate \(4pq\):
\[ 4 \times 14 = 56 \]

### Step 4: Substitute back into the equation
Now substitute this back into the equation:
\[ (p + q)^2 - 25 = 56 \]

### Step 5: Solve for \((p + q)^2\)
Add 25 to both sides:
\[ (p + q)^2 = 56 + 25 \]
\[ (p + q)^2 = 81 \]

### Conclusion
Thus, the value of \((p + q)^2\) is \(81\).

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