There are two quadratic equations. Solve these equations and find the relation between p and q.
`p^2-5p+4=0`
`q^2+5q-6=0`

Options

A. `ple q`
B. `pgtq`
C. `pge 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 \( p^2 - 5p + 4 = 0 \)

We can factor this equation. We need two numbers that multiply to \( 4 \) (the constant term) and add up to \( -5 \) (the coefficient of \( p \)).

The factors of \( 4 \) that satisfy this condition are \( -4 \) and \( -1 \).

So we can write:
\[ p^2 - 5p + 4 = (p - 4)(p - 1) = 0 \]

### Step 2: Find the values of \( p \)

Setting each factor to zero gives us:
1. \( p - 4 = 0 \) → \( p = 4 \)
2. \( p - 1 = 0 \) → \( p = 1 \)

Thus, the solutions for \( p \) are:
\[ p = 4 \quad \text{and} \quad p = 1 \]

### Step 3: Solve the second quadratic equation \( q^2 + 5q - 6 = 0 \)

Similarly, we can factor this equation. We need two numbers that multiply to \( -6 \) (the constant term) and add up to \( 5 \) (the coefficient of \( q \)).

The factors of \( -6 \) that satisfy this condition are \( 6 \) and \( -1 \).

So we can write:
\[ q^2 + 5q - 6 = (q + 6)(q - 1) = 0 \]

### Step 4: Find the values of \( q \)

Setting each factor to zero gives us:
1. \( q + 6 = 0 \) → \( q = -6 \)
2. \( q - 1 = 0 \) → \( q = 1 \)

Thus, the solutions for \( q \) are:
\[ q = -6 \quad \text{and} \quad q = 1 \]

### Step 5: Find the relation between \( p \) and \( q \)

Now we have the values:
- For \( p \): \( 4 \) and \( 1 \)
- For \( q \): \( -6 \) and \( 1 \)

We can compare these values:
- \( p = 4 \) is greater than both values of \( q \) (since \( 4 > -6 \) and \( 4 > 1 \)).
- \( p = 1 \) is equal to one value of \( q \) (since \( 1 = 1 \)) but greater than the other value of \( q \) (since \( 1 > -6 \)).

Thus, we can conclude:
\[ p > q \quad \text{for } p = 4 \quad \text{and} \quad p \geq q \quad \text{for } p = 1 \]

### Final Relation
The overall relation is:
\[ p \geq q \]

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