I. ` 2p^(2) + 20p+50 = 0`
II. ` q^(2) = 25`

Options

A. if`p lt q,`
B. if`p gt q,`
C. if` p le q,`
D. if` p ge q,`

Explanation

To solve the equations provided, we will approach them step by step.

### Step 1: Solve the first equation for P
The first equation is:
\[ 2p^2 + 20p + 50 = 0 \]

To simplify, we can divide the entire equation by 2:
\[ p^2 + 10p + 25 = 0 \]

Next, we can factor the quadratic equation. We need to find two numbers that multiply to 25 (the constant term) and add up to 10 (the coefficient of p). The numbers 5 and 5 work:
\[ (p + 5)(p + 5) = 0 \]
or
\[ (p + 5)^2 = 0 \]

Setting the factor equal to zero gives:
\[ p + 5 = 0 \]
So,
\[ p = -5 \]

### Step 2: Solve the second equation for Q
The second equation is:
\[ q^2 = 25 \]

Taking the square root of both sides, we find:
\[ q = \pm 5 \]
This means:
\[ q = 5 \quad \text{or} \quad q = -5 \]

### Step 3: Determine the relationship between P and Q
We have found:
- \( p = -5 \)
- \( q = 5 \) or \( q = -5 \)

Now, let's analyze the relationship:
- If \( q = 5 \), then \( q > p \) (since \( 5 > -5 \)).
- If \( q = -5 \), then \( q = p \) (since both are equal).

Thus, we can conclude that:
\[ q \geq p \]

### Final Answer
The relationship between P and Q is that \( q \) is greater than or equal to \( p \).

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