There are two quadratic equations. Solve these equations and find the relation between p and q.
`p^2-361=0`
`q^2-40q+399=0`

Options

A. `pltq`
B. `pgtq`
C. `pleqq`
D. `pleqq`

Answer

To solve the given quadratic equations and find the relation between \( p \) and \( q \), we will follow these steps:

### Step 1: Solve the first quadratic equation \( p^2 - 361 = 0 \)

1. Rewrite the equation:
\[ p^2 - 361 = 0 \]

2. Recognize that \( 361 \) is a perfect square:
\[ p^2 - 19^2 = 0 \]

3. Factor the equation using the difference of squares:
\[ (p + 19)(p - 19) = 0 \]

4. Set each factor to zero:
\[ p + 19 = 0 \quad \text{or} \quad p - 19 = 0 \]

5. Solve for \( p \):
\[ p = -19 \quad \text{or} \quad p = 19 \]

### Step 2: Solve the second quadratic equation \( q^2 - 40q + 399 = 0 \)

1. Rewrite the equation:
\[ q^2 - 40q + 399 = 0 \]

2. We will use the factorization method. We need to find two numbers that multiply to \( 399 \) and add up to \( -40 \). These numbers are \( -21 \) and \( -19 \).

3. Rewrite the equation:
\[ q^2 - 21q - 19q + 399 = 0 \]

4. Factor by grouping:
\[ q(q - 21) - 19(q - 21) = 0 \]
\[ (q - 21)(q - 19) = 0 \]

5. Set each factor to zero:
\[ q - 21 = 0 \quad \text{or} \quad q - 19 = 0 \]

6. Solve for \( q \):
\[ q = 21 \quad \text{or} \quad q = 19 \]

### Step 3: Find the relation between \( p \) and \( q \)

1. We have the values:
- \( p = -19 \) or \( p = 19 \)
- \( q = 21 \) or \( q = 19 \)

2. Compare the values:
- For \( p = 19 \):
- \( q = 19 \) gives \( p = q \)
- \( q = 21 \) gives \( p < q \)
- For \( p = -19 \):
- \( q = 19 \) gives \( p < q \)
- \( q = 21 \) gives \( p < q \)

### Conclusion

From the comparisons, we can conclude that:
\[ p \leq q \]