p and q are the zeroes of the polynomial `4y^(2) - 4y+1`. What is the value of `(1)/(p) + (1)/(q) + pq `?

Options

A. `(-15)/(4)`
B. `(-3)/(4)`
C. `(5)/(4)`
D. `(17)/(4)`

Answer

To solve the problem, we need to find the value of \( \frac{1}{p} + \frac{1}{q} + pq \) where \( p \) and \( q \) are the zeros of the polynomial \( 4y^2 - 4y + 1 \).

### Step 1: Identify the coefficients
The given polynomial is \( 4y^2 - 4y + 1 \). Here, we can identify:
- \( A = 4 \)
- \( B = -4 \)
- \( C = 1 \)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( p + q \) is given by:
\[ p + q = -\frac{B}{A} = -\frac{-4}{4} = 1 \]
- The product of the roots \( pq \) is given by:
\[ pq = \frac{C}{A} = \frac{1}{4} \]

### Step 3: Substitute into the expression
We need to find the value of \( \frac{1}{p} + \frac{1}{q} + pq \). We can rewrite \( \frac{1}{p} + \frac{1}{q} \) as:
\[ \frac{1}{p} + \frac{1}{q} = \frac{p + q}{pq} \]
Substituting the values we found:
\[ \frac{1}{p} + \frac{1}{q} = \frac{1}{\frac{1}{4}} = 4 \]

### Step 4: Add \( pq \) to the expression
Now we add \( pq \) to \( \frac{1}{p} + \frac{1}{q} \):
\[ \frac{1}{p} + \frac{1}{q} + pq = 4 + \frac{1}{4} \]

### Step 5: Combine the terms
To combine \( 4 \) and \( \frac{1}{4} \), we can express \( 4 \) as \( \frac{16}{4} \):
\[ 4 + \frac{1}{4} = \frac{16}{4} + \frac{1}{4} = \frac{17}{4} \]

### Final Answer
Thus, the value of \( \frac{1}{p} + \frac{1}{q} + pq \) is:
\[ \frac{17}{4} \]