If p and q are positive, `p^2 + q^2 = 16 , and p^2 - q^2 = 8`, then q =

Options

A. 2
B. 4
C. 8
D. `2sqrt(2)`

Answer

To solve the equations \( p^2 + q^2 = 16 \) and \( p^2 - q^2 = 8 \), we can follow these steps:

### Step 1: Write down the equations
We have two equations:
1. \( p^2 + q^2 = 16 \) (Equation 1)
2. \( p^2 - q^2 = 8 \) (Equation 2)

### Step 2: Add the two equations
Now, we will add Equation 1 and Equation 2:
\[ (p^2 + q^2) + (p^2 - q^2) = 16 + 8 \]
This simplifies to:
\[ 2p^2 = 24 \]

### Step 3: Solve for \( p^2 \)
Now, divide both sides by 2:
\[ p^2 = \frac{24}{2} = 12 \]

### Step 4: Substitute \( p^2 \) back into Equation 1
Next, we will substitute \( p^2 \) back into Equation 1 to find \( q^2 \):
\[ 12 + q^2 = 16 \]

### Step 5: Solve for \( q^2 \)
Now, isolate \( q^2 \):
\[ q^2 = 16 - 12 = 4 \]

### Step 6: Find \( q \)
Finally, take the square root of both sides to find \( q \):
\[ q = \sqrt{4} = 2 \]

Thus, the value of \( q \) is \( 2 \).

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