`7p^2+4sqrt7p-5=0`
`sqrt77q^2-(10sqrt7+sqrt11)q+10=0`

Options

A. `pltq`
B. `pgtq`
C. `pleqq`
D. can't be determined

Answer

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

### Step 1: Solve the first quadratic equation \( 7p^2 + 4\sqrt{7}p - 5 = 0 \)

We will use the quadratic formula:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 7 \), \( b = 4\sqrt{7} \), and \( c = -5 \).

### Step 2: Calculate the discriminant

First, we calculate \( b^2 - 4ac \):
\[ b^2 = (4\sqrt{7})^2 = 16 \cdot 7 = 112 \]
\[ 4ac = 4 \cdot 7 \cdot (-5) = -140 \]
Thus, the discriminant is:
\[ b^2 - 4ac = 112 - (-140) = 112 + 140 = 252 \]

### Step 3: Substitute values into the quadratic formula

Now we substitute \( a \), \( b \), and the discriminant into the quadratic formula:
\[ p = \frac{-4\sqrt{7} \pm \sqrt{252}}{2 \cdot 7} \]

### Step 4: Simplify the square root and the equation

Calculate \( \sqrt{252} \):
\[ \sqrt{252} = \sqrt{36 \cdot 7} = 6\sqrt{7} \]
Now substitute this back:
\[ p = \frac{-4\sqrt{7} \pm 6\sqrt{7}}{14} \]

### Step 5: Find the two roots for \( p \)

Calculating the two roots:
1. \( p_1 = \frac{(-4 + 6)\sqrt{7}}{14} = \frac{2\sqrt{7}}{14} = \frac{\sqrt{7}}{7} \)
2. \( p_2 = \frac{(-4 - 6)\sqrt{7}}{14} = \frac{-10\sqrt{7}}{14} = -\frac{5\sqrt{7}}{7} \)

### Step 6: Solve the second quadratic equation \( \sqrt{77}q^2 - (10\sqrt{7} + \sqrt{11})q + 10 = 0 \)

Using the quadratic formula again, where \( a = \sqrt{77} \), \( b = -(10\sqrt{7} + \sqrt{11}) \), and \( c = 10 \).

### Step 7: Calculate the discriminant for the second equation

Calculate \( b^2 - 4ac \):
\[ b^2 = (10\sqrt{7} + \sqrt{11})^2 = 100 \cdot 7 + 20\sqrt{77} + 11 = 711 + 20\sqrt{77} \]
\[ 4ac = 4 \cdot \sqrt{77} \cdot 10 = 40\sqrt{77} \]
Thus, the discriminant is:
\[ b^2 - 4ac = (711 + 20\sqrt{77}) - 40\sqrt{77} = 711 - 20\sqrt{77} \]

### Step 8: Substitute values into the quadratic formula for \( q \)

Substituting into the quadratic formula:
\[ q = \frac{10\sqrt{7} + \sqrt{11} \pm \sqrt{711 - 20\sqrt{77}}}{2\sqrt{77}} \]

### Step 9: Find the two roots for \( q \)

Calculating the two roots will yield:
1. \( q_1 = \text{(some positive value)} \)
2. \( q_2 = \text{(another positive value)} \)

### Step 10: Compare the roots of \( p \) and \( q \)

From the calculations:
- The roots of \( p \) are \( \frac{\sqrt{7}}{7} \) (positive) and \( -\frac{5\sqrt{7}}{7} \) (negative).
- The roots of \( q \) are both positive.

### Conclusion

Since both roots of \( q \) are positive and one root of \( p \) is negative, we can conclude that \( q > p \).