There are two quadratic equations. Solve these equations and find the relation between p and q.
`p^2-8p+15=0`
`q^2+8q+12=0`

Options

A. `pltq`
B. `pgtq`
C. `ple q`
D. connot be determined

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 \( p^2 - 8p + 15 = 0 \)

We can factor the quadratic equation:

\[ p^2 - 8p + 15 = (p - 5)(p - 3) = 0 \]

Setting each factor to zero gives us:

\[ p - 5 = 0 \quad \Rightarrow \quad p = 5 \]
\[ p - 3 = 0 \quad \Rightarrow \quad p = 3 \]

Thus, the solutions for \( p \) are:

\[ p = 5 \quad \text{or} \quad p = 3 \]

### Step 2: Solve the second quadratic equation \( q^2 + 8q + 12 = 0 \)

We can factor this quadratic equation as well:

\[ q^2 + 8q + 12 = (q + 6)(q + 2) = 0 \]

Setting each factor to zero gives us:

\[ q + 6 = 0 \quad \Rightarrow \quad q = -6 \]
\[ q + 2 = 0 \quad \Rightarrow \quad q = -2 \]

Thus, the solutions for \( q \) are:

\[ q = -6 \quad \text{or} \quad q = -2 \]

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

Now we have the values:

- \( p = 5 \) or \( p = 3 \)
- \( q = -6 \) or \( q = -2 \)

To establish a relation, we can observe the values:

1. If \( p = 5 \), we can relate it to \( q \):
\[ p + q = 5 - 6 = -1 \quad \text{or} \quad p + q = 5 - 2 = 3 \]

2. If \( p = 3 \), we can relate it to \( q \):
\[ p + q = 3 - 6 = -3 \quad \text{or} \quad p + q = 3 - 2 = 1 \]

From these calculations, we can summarize the relations as follows:

- For \( p = 5 \), \( q \) can take values that yield \( p + q = -1 \) or \( p + q = 3 \).
- For \( p = 3 \), \( q \) can yield \( p + q = -3 \) or \( p + q = 1 \).

### Conclusion

The relation between \( p \) and \( q \) can be summarized as:

- \( p + q = -1, 3, -3, 1 \)