Direction: In the following questions two equations numbered (I) and (II) are given. You have to solve both equations and Give answer
I. `2p^2+p-28=0` II. `2q^2-23q+56=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 start with each equation separately.

### Step 1: Solve Equation I
The first equation is:
\[ 2p^2 + p - 28 = 0 \]

To solve this quadratic equation, we can factor it. We need to find two numbers that multiply to \(2 \times -28 = -56\) and add to \(1\) (the coefficient of \(p\)).

The factors of \(-56\) that add up to \(1\) are \(8\) and \(-7\).

So, we can rewrite the equation as:
\[ 2p^2 + 8p - 7p - 28 = 0 \]

Now, we can group the terms:
\[ (2p^2 + 8p) + (-7p - 28) = 0 \]

Factoring out the common terms:
\[ 2p(p + 4) - 7(p + 4) = 0 \]

Now, we can factor out \((p + 4)\):
\[ (2p - 7)(p + 4) = 0 \]

Setting each factor to zero gives us:
1. \(2p - 7 = 0 \Rightarrow p = \frac{7}{2}\)
2. \(p + 4 = 0 \Rightarrow p = -4\)

### Step 2: Solve Equation II
The second equation is:
\[ 2q^2 - 23q + 56 = 0 \]

Again, we will factor this quadratic equation. We need two numbers that multiply to \(2 \times 56 = 112\) and add to \(-23\).

The factors of \(112\) that add up to \(-23\) are \(-16\) and \(-7\).

So, we can rewrite the equation as:
\[ 2q^2 - 16q - 7q + 56 = 0 \]

Now, we can group the terms:
\[ (2q^2 - 16q) + (-7q + 56) = 0 \]

Factoring out the common terms:
\[ 2q(q - 8) - 7(q - 8) = 0 \]

Now, we can factor out \((q - 8)\):
\[ (2q - 7)(q - 8) = 0 \]

Setting each factor to zero gives us:
1. \(2q - 7 = 0 \Rightarrow q = \frac{7}{2}\)
2. \(q - 8 = 0 \Rightarrow q = 8\)

### Step 3: Compare Values of p and q
Now we have the values:
- For \(p\): \(\frac{7}{2}\) and \(-4\)
- For \(q\): \(\frac{7}{2}\) and \(8\)

We will compare the values of \(p\) and \(q\):

1. When \(p = \frac{7}{2}\):
- \(q = \frac{7}{2}\) → \(p = q\)
- \(q = 8\) → \(p < q\)

2. When \(p = -4\):
- \(q = \frac{7}{2}\) → \(p < q\)
- \(q = 8\) → \(p < q\)

### Conclusion
From the comparisons, we can conclude:
- \(p\) is equal to \(q\) in one case.
- In both cases, \(p\) is less than \(q\).

Thus, we can say:
\[ q \geq p \]

### Final Answer
The correct option is:
\[ q \text{ is greater than or equal to } p \]