Find all the parameters p for which both the roots of the equation `x^2-6px+(3-4p+9p^2)=0` exceed the number 4.

Options

A. `pgt3/4`
B. `pgt4/3`
C. `plt1`
D. `pgt13/9`

Answer

To find the parameters \( p \) for which both roots of the equation \( x^2 - 6px + (3 - 4p + 9p^2) = 0 \) exceed the number 4, we can follow these steps:

### Step 1: Set the condition for the roots
We want both roots of the quadratic equation to be greater than 4. For this, we can use the fact that if \( x = 4 \) is a root, we can derive conditions on \( p \).

### Step 2: Substitute \( x = 4 \) into the equation
Substituting \( x = 4 \) into the equation gives:
\[ 4^2 - 6p(4) + (3 - 4p + 9p^2) = 0 \]
This simplifies to:
\[ 16 - 24p + 3 - 4p + 9p^2 = 0 \]
Combining like terms, we have:
\[ 9p^2 - 28p + 19 = 0 \]

### Step 3: Find the conditions for the roots
For both roots to be greater than 4, we need to ensure two conditions:
1. The sum of the roots \( \frac{-b}{a} = \frac{6p}{1} > 8 \) (since \( 4 + 4 = 8 \)).
2. The product of the roots \( \frac{c}{a} = \frac{3 - 4p + 9p^2}{1} > 16 \).

### Step 4: Solve the first condition
From the first condition:
\[ 6p > 8 \implies p > \frac{8}{6} \implies p > \frac{4}{3} \]

### Step 5: Solve the second condition
From the second condition:
\[ 3 - 4p + 9p^2 > 16 \]
This simplifies to:
\[ 9p^2 - 4p - 13 > 0 \]
We can factor this quadratic inequality. First, we find the roots of the equation \( 9p^2 - 4p - 13 = 0 \) using the quadratic formula:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 9 \cdot (-13)}}{2 \cdot 9} \]
Calculating the discriminant:
\[ = \frac{4 \pm \sqrt{16 + 468}}{18} = \frac{4 \pm \sqrt{484}}{18} = \frac{4 \pm 22}{18} \]
This gives us:
\[ p = \frac{26}{18} = \frac{13}{9} \quad \text{and} \quad p = \frac{-18}{18} = -1 \]

### Step 6: Analyze the intervals
The quadratic \( 9p^2 - 4p - 13 \) opens upwards (since the coefficient of \( p^2 \) is positive). Therefore, the inequality \( 9p^2 - 4p - 13 > 0 \) holds outside the roots:
1. \( p < -1 \)
2. \( p > \frac{13}{9} \)

### Step 7: Combine the conditions
We have two conditions:
1. \( p > \frac{4}{3} \) (which is approximately \( 1.33 \))
2. \( p < -1 \) or \( p > \frac{13}{9} \) (which is approximately \( 1.44 \))

Since \( p < -1 \) does not satisfy \( p > \frac{4}{3} \), we only consider \( p > \frac{13}{9} \).

### Final Result
Thus, the values of \( p \) for which both roots of the equation exceed 4 are:
\[ p > \frac{13}{9} \]