There are two quadratic equations. Solve these equations and find the relation Between p and q.
`3p^2+5p+5=0`
`q^2-4q-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 \( 3p^2 + 5p + 5 = 0 \)

We will use the quadratic formula:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 3 \), \( b = 5 \), and \( c = 5 \).

#### Calculation:
1. Calculate the discriminant \( D = b^2 - 4ac \):
\[ D = 5^2 - 4 \cdot 3 \cdot 5 = 25 - 60 = -35 \]
2. Since the discriminant is negative, the roots will be complex (imaginary).

Thus, we can express the roots as:
\[ p = \frac{-5 \pm \sqrt{-35}}{6} = \frac{-5 \pm i\sqrt{35}}{6} \]

### Step 2: Solve the second quadratic equation \( q^2 - 4q - 12 = 0 \)

Again, we will use the quadratic formula:
\[ q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -4 \), and \( c = -12 \).

#### Calculation:
1. Calculate the discriminant \( D = b^2 - 4ac \):
\[ D = (-4)^2 - 4 \cdot 1 \cdot (-12) = 16 + 48 = 64 \]
2. Since the discriminant is positive, we can find real roots:
\[ q = \frac{4 \pm \sqrt{64}}{2} = \frac{4 \pm 8}{2} \]
3. Thus, the roots are:
- \( q = \frac{12}{2} = 6 \)
- \( q = \frac{-4}{2} = -2 \)

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

From the first equation, we found that \( p \) has complex roots:
\[ p = \frac{-5 \pm i\sqrt{35}}{6} \]
From the second equation, we found that \( q \) has real roots:
\[ q = 6 \quad \text{or} \quad q = -2 \]

### Conclusion

Since \( p \) has complex roots and \( q \) has real roots, there is no direct relation between \( p \) and \( q \) in terms of equality or a simple function. The roots of the first equation do not yield any real values that can relate to the real roots of the second equation.

### Final Answer
The relation between \( p \) and \( q \) cannot be determined due to the nature of their roots.

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