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^2-5p+6=0`
`q^2-4q+3=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 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:
\[ p^2 - 5p + 6 = 0 \]
We can factor this equation. We look for two numbers that multiply to \( 6 \) (the constant term) and add to \( -5 \) (the coefficient of \( p \)). The numbers \( -2 \) and \( -3 \) satisfy these conditions.

Thus, we can factor the equation as:
\[ (p - 2)(p - 3) = 0 \]
Setting each factor to zero gives us the solutions:
\[ p - 2 = 0 \quad \Rightarrow \quad p = 2 \]
\[ p - 3 = 0 \quad \Rightarrow \quad p = 3 \]
So, the possible values of \( p \) are \( 2 \) and \( 3 \).

### Step 2: Solve the second equation for \( q \)
The second equation is:
\[ q^2 - 4q + 3 = 0 \]
Again, we look for two numbers that multiply to \( 3 \) and add to \( -4 \). The numbers \( -1 \) and \( -3 \) work here.

Thus, we can factor the equation as:
\[ (q - 1)(q - 3) = 0 \]
Setting each factor to zero gives us the solutions:
\[ q - 1 = 0 \quad \Rightarrow \quad q = 1 \]
\[ q - 3 = 0 \quad \Rightarrow \quad q = 3 \]
So, the possible values of \( q \) are \( 1 \) and \( 3 \).

### Step 3: Determine the relationship between \( p \) and \( q \)
Now we have the possible values:
- For \( p \): \( 2, 3 \)
- For \( q \): \( 1, 3 \)

We will compare the values of \( p \) and \( q \):
1. If \( p = 2 \):
- \( q = 1 \) → \( p > q \)
- \( q = 3 \) → \( p < q \)

2. If \( p = 3 \):
- \( q = 1 \) → \( p > q \)
- \( q = 3 \) → \( p = q \)

From these comparisons, we can conclude:
- \( p \) can be greater than, less than, or equal to \( q \) depending on the specific values chosen.

### Final Conclusion
Since there are cases where \( p > q \), \( p < q \), and \( p = q \), we cannot definitively establish a single relationship between \( p \) and \( q \). Therefore, the answer is that no relation can be decided between \( p \) and \( q \).

---