If the quadratic equation `2x^(2)+5x+1=0` has roots p and q,what is the value of the expression (2-p)(2-q)?

Options

A. 0
B. -2
C. `sqrt(17)`
D. 9.5

Answer

To solve the problem, we need to find the value of the expression \( (2 - p)(2 - q) \) where \( p \) and \( q \) are the roots of the quadratic equation \( 2x^2 + 5x + 1 = 0 \).

### Step 1: Identify the coefficients
The given quadratic equation is \( 2x^2 + 5x + 1 = 0 \). Here, the coefficients are:
- \( a = 2 \)
- \( b = 5 \)
- \( c = 1 \)

### Step 2: Use the quadratic formula to find the roots
The roots \( p \) and \( q \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} \]
Calculating the discriminant:
\[ b^2 - 4ac = 25 - 8 = 17 \]
So the roots are:
\[ x = \frac{-5 \pm \sqrt{17}}{4} \]
Thus, we have:
\[ p = \frac{-5 + \sqrt{17}}{4}, \quad q = \frac{-5 - \sqrt{17}}{4} \]

### Step 3: Calculate \( 2 - p \) and \( 2 - q \)
Now, we calculate \( 2 - p \) and \( 2 - q \):
\[ 2 - p = 2 - \frac{-5 + \sqrt{17}}{4} = 2 + \frac{5 - \sqrt{17}}{4} = \frac{8 + 5 - \sqrt{17}}{4} = \frac{13 - \sqrt{17}}{4} \]
\[ 2 - q = 2 - \frac{-5 - \sqrt{17}}{4} = 2 + \frac{5 + \sqrt{17}}{4} = \frac{8 + 5 + \sqrt{17}}{4} = \frac{13 + \sqrt{17}}{4} \]

### Step 4: Multiply \( (2 - p)(2 - q) \)
Now we need to multiply \( (2 - p)(2 - q) \):
\[ (2 - p)(2 - q) = \left(\frac{13 - \sqrt{17}}{4}\right)\left(\frac{13 + \sqrt{17}}{4}\right) \]
Using the difference of squares:
\[ = \frac{(13 - \sqrt{17})(13 + \sqrt{17})}{16} = \frac{13^2 - (\sqrt{17})^2}{16} = \frac{169 - 17}{16} = \frac{152}{16} = \frac{38}{4} = 9.5 \]

### Final Answer
Thus, the value of the expression \( (2 - p)(2 - q) \) is \( 9.5 \).

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