Direction: In the following questions two equations numbered (I) and (II) are given. You have to solve both equations and Give answer
I. `p^2-8p+15=0` II. `2q^2-7q+5=0`

Options

A. If `p gt q`
B. If `p ge q`
C. If `q gt p`
D. if `q ge p`

Answer

To solve the given equations step by step, we will tackle each equation separately.

### Step 1: Solve Equation I
The first equation is:
\[ p^2 - 8p + 15 = 0 \]

To factor this quadratic equation, we need to find two numbers that multiply to \(15\) (the constant term) and add up to \(-8\) (the coefficient of \(p\)).

The numbers that satisfy these conditions are \(-5\) and \(-3\).

Thus, we can factor the equation as:
\[ (p - 5)(p - 3) = 0 \]

### Step 2: Find the Roots for Equation I
Setting each factor to zero gives us:
1. \( p - 5 = 0 \) → \( p = 5 \)
2. \( p - 3 = 0 \) → \( p = 3 \)

So, the solutions for \(p\) are:
\[ p = 3 \quad \text{and} \quad p = 5 \]

### Step 3: Solve Equation II
The second equation is:
\[ 2q^2 - 7q + 5 = 0 \]

To factor this quadratic equation, we need to find two numbers that multiply to \(2 \times 5 = 10\) and add up to \(-7\).

The numbers that satisfy these conditions are \(-5\) and \(-2\).

Thus, we can rewrite the equation as:
\[ 2q^2 - 5q - 2q + 5 = 0 \]
Grouping the terms gives us:
\[ q(2q - 5) - 1(2q - 5) = 0 \]
Factoring out \( (2q - 5) \) gives us:
\[ (2q - 5)(q - 1) = 0 \]

### Step 4: Find the Roots for Equation II
Setting each factor to zero gives us:
1. \( 2q - 5 = 0 \) → \( q = \frac{5}{2} \)
2. \( q - 1 = 0 \) → \( q = 1 \)

So, the solutions for \(q\) are:
\[ q = 1 \quad \text{and} \quad q = \frac{5}{2} \]

### Step 5: Compare the Values of p and q
Now we have the following possible pairs of \(p\) and \(q\):
1. \( p = 3 \) and \( q = 1 \)
2. \( p = 3 \) and \( q = \frac{5}{2} \)
3. \( p = 5 \) and \( q = 1 \)
4. \( p = 5 \) and \( q = \frac{5}{2} \)

Now we will compare \(p\) and \(q\) in each case:
1. \( 3 > 1 \) (True)
2. \( 3 > \frac{5}{2} \) (True)
3. \( 5 > 1 \) (True)
4. \( 5 > \frac{5}{2} \) (True)

In all cases, \(p\) is greater than \(q\).

### Conclusion
Thus, we conclude that \(p\) is greater than \(q\).

### Final Answer
The correct answer is: **p is greater than q**.

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