If 3 - 4i is a root of `x^(2) - px + q = 0 ` where p, q `in R`. then value `(2 p - q)/( p + q)` is

Options

A. `-12//31`
B. `-13//31`
C. `-15//31`
D. none of these

Answer

To solve the problem, we start with the quadratic equation given:

\[ x^2 - px + q = 0 \]

We know that one of the roots is \( 3 - 4i \). Since the coefficients \( p \) and \( q \) are real numbers, the other root must be the conjugate of the first root, which is \( 3 + 4i \).

### Step 1: Find the sum of the roots
The sum of the roots \( r_1 \) and \( r_2 \) of the quadratic equation can be expressed as:

\[ r_1 + r_2 = (3 - 4i) + (3 + 4i) = 6 \]

According to Vieta's formulas, the sum of the roots is also given by:

\[ r_1 + r_2 = \frac{p}{1} \]

Thus, we have:

\[ p = 6 \]

### Step 2: Find the product of the roots
Now, we calculate the product of the roots:

\[ r_1 \cdot r_2 = (3 - 4i)(3 + 4i) = 3^2 - (4i)^2 = 9 - (-16) = 9 + 16 = 25 \]

Again, according to Vieta's formulas, the product of the roots is given by:

\[ r_1 \cdot r_2 = q \]

Thus, we have:

\[ q = 25 \]

### Step 3: Calculate \( \frac{2p - q}{p + q} \)
Now we can substitute the values of \( p \) and \( q \) into the expression \( \frac{2p - q}{p + q} \):

\[ 2p - q = 2(6) - 25 = 12 - 25 = -13 \]

\[ p + q = 6 + 25 = 31 \]

Now substituting these into the expression:

\[ \frac{2p - q}{p + q} = \frac{-13}{31} \]

### Final Answer
Thus, the value of \( \frac{2p - q}{p + q} \) is:

\[ \frac{-13}{31} \]

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