`225p^2-1740p+2835=0`
`225q^2+690q-2835=0`

Options

A. `pltq`
B. `pgtq`
C. `pleqq`
D. `pgeqq`

Answer

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

### Step 1: Simplify the first equation
The first equation is:
\[ 225p^2 - 1740p + 2835 = 0 \]
We can factor out 15 from the equation:
\[ 15(15p^2 - 116p + 189) = 0 \]
Thus, we simplify it to:
\[ 15p^2 - 116p + 189 = 0 \]

### Step 2: Apply the quadratic formula for \( p \)
The quadratic formula is given by:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation, \( a = 15 \), \( b = -116 \), and \( c = 189 \).

Calculating \( b^2 - 4ac \):
\[ b^2 = (-116)^2 = 13456 \]
\[ 4ac = 4 \times 15 \times 189 = 11340 \]
Thus,
\[ b^2 - 4ac = 13456 - 11340 = 2116 \]

Now substituting into the quadratic formula:
\[ p = \frac{116 \pm \sqrt{2116}}{30} \]
Calculating \( \sqrt{2116} = 46 \):
\[ p = \frac{116 \pm 46}{30} \]

Calculating the two possible values for \( p \):
1. \( p_1 = \frac{116 + 46}{30} = \frac{162}{30} = \frac{27}{5} \)
2. \( p_2 = \frac{116 - 46}{30} = \frac{70}{30} = \frac{7}{3} \)

### Step 3: Simplify the second equation
The second equation is:
\[ 225q^2 + 690q - 2835 = 0 \]
We can factor out 15 from the equation:
\[ 15(15q^2 + 46q - 189) = 0 \]
Thus, we simplify it to:
\[ 15q^2 + 46q - 189 = 0 \]

### Step 4: Apply the quadratic formula for \( q \)
Using the quadratic formula again, where \( a = 15 \), \( b = 46 \), and \( c = -189 \).

Calculating \( b^2 - 4ac \):
\[ b^2 = 46^2 = 2116 \]
\[ 4ac = 4 \times 15 \times (-189) = -11340 \]
Thus,
\[ b^2 - 4ac = 2116 + 11340 = 13456 \]

Now substituting into the quadratic formula:
\[ q = \frac{-46 \pm \sqrt{13456}}{30} \]
Calculating \( \sqrt{13456} = 116 \):
\[ q = \frac{-46 \pm 116}{30} \]

Calculating the two possible values for \( q \):
1. \( q_1 = \frac{-46 + 116}{30} = \frac{70}{30} = \frac{7}{3} \)
2. \( q_2 = \frac{-46 - 116}{30} = \frac{-162}{30} = -\frac{27}{5} \)

### Step 5: Compare the values of \( p \) and \( q \)
We have:
- \( p_1 = \frac{27}{5} \), \( p_2 = \frac{7}{3} \)
- \( q_1 = \frac{7}{3} \), \( q_2 = -\frac{27}{5} \)

Now, we can compare:
- \( \frac{7}{3} \) (for \( p_2 \)) is equal to \( \frac{7}{3} \) (for \( q_1 \)).
- \( \frac{27}{5} \) (for \( p_1 \)) is greater than \( -\frac{27}{5} \) (for \( q_2 \)).

Thus, we conclude that:
\[ p \geq q \]

### Final Answer
The relationship between \( p \) and \( q \) is:
\[ p \geq q \]