If `x^(2) + px + q = 0` is the quadratic equation whose roots are a-2 and b-2 where a and b are the roots of `x^(2)-3x + 1 = 0` then

Options

A. p = 1, q = 5
B. p = 1, q = -5
C. p = 1, q = 2
D. p = 1, q = -1

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) in the quadratic equation \( x^2 + px + q = 0 \), where the roots are \( a - 2 \) and \( b - 2 \). Here, \( a \) and \( b \) are the roots of the equation \( x^2 - 3x + 1 = 0 \).

### Step 1: Find the roots \( a \) and \( b \) of the equation \( x^2 - 3x + 1 = 0 \).

Using the quadratic formula:
\[ a, b = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation, \( a = 1 \), \( b = -3 \), and \( c = 1 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 1 = 9 - 4 = 5 \]

Now substituting into the quadratic formula:
\[ a, b = \frac{3 \pm \sqrt{5}}{2} \]

### Step 2: Calculate the roots \( a \) and \( b \).

Thus, the roots are:
\[ a = \frac{3 + \sqrt{5}}{2}, \quad b = \frac{3 - \sqrt{5}}{2} \]

### Step 3: Find the new roots \( a - 2 \) and \( b - 2 \).

Calculating \( a - 2 \):
\[ a - 2 = \frac{3 + \sqrt{5}}{2} - 2 = \frac{3 + \sqrt{5} - 4}{2} = \frac{\sqrt{5} - 1}{2} \]

Calculating \( b - 2 \):
\[ b - 2 = \frac{3 - \sqrt{5}}{2} - 2 = \frac{3 - \sqrt{5} - 4}{2} = \frac{-\sqrt{5} - 1}{2} \]

### Step 4: Use the new roots to find \( p \) and \( q \).

The sum of the roots \( (a - 2) + (b - 2) \):
\[ \left( \frac{\sqrt{5} - 1}{2} \right) + \left( \frac{-\sqrt{5} - 1}{2} \right) = \frac{\sqrt{5} - 1 - \sqrt{5} - 1}{2} = \frac{-2}{2} = -1 \]

Thus, \( p = -\text{(sum of roots)} = -(-1) = 1 \).

The product of the roots \( (a - 2)(b - 2) \):
\[ \left( \frac{\sqrt{5} - 1}{2} \right) \left( \frac{-\sqrt{5} - 1}{2} \right) = \frac{(\sqrt{5} - 1)(-\sqrt{5} - 1)}{4} \]
Expanding this:
\[ = \frac{-5 - \sqrt{5} + \sqrt{5} + 1}{4} = \frac{-4}{4} = -1 \]

Thus, \( q = \text{(product of roots)} = -1 \).

### Final Answer:
The values of \( p \) and \( q \) are:
\[ p = 1, \quad q = -1 \]

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