Two equations 1 and 2 are given in each question. On the basis of these equations you have to decide the relation between `p` and `q` and give answer.
`p^2+13p+40=0`
`q^2+7q+12=0`

Options

A. If no relation can be decided between `p` and `q`
B. If `p gt q`
C. If `q gt p`
D. If `p=q`

Answer

To solve the given equations and determine the relationship between \( p \) and \( q \), we will follow these steps:

### Step 1: Solve the first equation for \( p \)
The first equation is:
\[ p^2 + 13p + 40 = 0 \]
We can factor this quadratic equation. We need to find two numbers that multiply to \( 40 \) (the constant term) and add up to \( 13 \) (the coefficient of \( p \)).

The numbers that satisfy these conditions are \( 8 \) and \( 5 \). Therefore, we can factor the equation as:
\[ (p + 8)(p + 5) = 0 \]
Setting each factor to zero gives us:
\[ p + 8 = 0 \quad \Rightarrow \quad p = -8 \]
\[ p + 5 = 0 \quad \Rightarrow \quad p = -5 \]
Thus, the solutions for \( p \) are:
\[ p = -8 \quad \text{and} \quad p = -5 \]

### Step 2: Solve the second equation for \( q \)
The second equation is:
\[ q^2 + 7q + 12 = 0 \]
Similarly, we will factor this quadratic equation. We need to find two numbers that multiply to \( 12 \) and add up to \( 7 \).

The numbers that satisfy these conditions are \( 4 \) and \( 3 \). Therefore, we can factor the equation as:
\[ (q + 4)(q + 3) = 0 \]
Setting each factor to zero gives us:
\[ q + 4 = 0 \quad \Rightarrow \quad q = -4 \]
\[ q + 3 = 0 \quad \Rightarrow \quad q = -3 \]
Thus, the solutions for \( q \) are:
\[ q = -4 \quad \text{and} \quad q = -3 \]

### Step 3: Compare the values of \( p \) and \( q \)
Now we have the values:
- For \( p \): \( -8 \) and \( -5 \)
- For \( q \): \( -4 \) and \( -3 \)

We will compare the maximum values of \( p \) and \( q \):
- The maximum value of \( p \) is \( -5 \).
- The maximum value of \( q \) is \( -3 \).

### Step 4: Determine the relationship
Since \( -5 < -3 \), we can conclude that:
\[ p < q \]

### Final Answer
The relationship between \( p \) and \( q \) is:
\[ p < q \]

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