The equation formed by increasing each roots of `2x^(2) - 3x - 1 = 0` by 2 is

Options

A. `2x^(2) + 5x + 1 = 0`
B. `2x^(2) + 5x + 3 = 0`
C. `2x^(2) -11x -18 = 0`
D. `2x^(2) -11x + 13 = 0`

Answer

To solve the problem, we need to find the equation formed by increasing each root of the given quadratic equation \(2x^2 - 3x - 1 = 0\) by 2.

### Step-by-Step Solution:

1. **Identify the given quadratic equation**:
The equation we are given is:
\[ 2x^2 - 3x - 1 = 0 \]

2. **Find the roots of the equation**:
We can use the quadratic formula to find the roots. The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 2\), \(b = -3\), and \(c = -1\).

3. **Calculate the discriminant**:
\[ b^2 - 4ac = (-3)^2 - 4 \cdot 2 \cdot (-1) = 9 + 8 = 17 \]

4. **Find the roots using the quadratic formula**:
\[ x = \frac{-(-3) \pm \sqrt{17}}{2 \cdot 2} = \frac{3 \pm \sqrt{17}}{4} \]
Thus, the roots are:
\[ \alpha = \frac{3 + \sqrt{17}}{4}, \quad \beta = \frac{3 - \sqrt{17}}{4} \]

5. **Increase each root by 2**:
The new roots will be:
\[ \alpha + 2 = \frac{3 + \sqrt{17}}{4} + 2 = \frac{3 + \sqrt{17}}{4} + \frac{8}{4} = \frac{11 + \sqrt{17}}{4} \]
\[ \beta + 2 = \frac{3 - \sqrt{17}}{4} + 2 = \frac{3 - \sqrt{17}}{4} + \frac{8}{4} = \frac{11 - \sqrt{17}}{4} \]

6. **Form the new quadratic equation**:
The sum of the new roots is:
\[ (\alpha + 2) + (\beta + 2) = \left(\frac{11 + \sqrt{17}}{4} + \frac{11 - \sqrt{17}}{4}\right) = \frac{22}{4} = \frac{11}{2} \]
The product of the new roots is:
\[ (\alpha + 2)(\beta + 2) = \left(\frac{11 + \sqrt{17}}{4}\right)\left(\frac{11 - \sqrt{17}}{4}\right) = \frac{(11)^2 - (\sqrt{17})^2}{16} = \frac{121 - 17}{16} = \frac{104}{16} = \frac{13}{2} \]

7. **Write the new quadratic equation**:
The new quadratic equation can be formed using the sum and product of the roots:
\[ x^2 - \left(\text{sum of roots}\right)x + \left(\text{product of roots}\right) = 0 \]
Substituting the values:
\[ x^2 - \frac{11}{2}x + \frac{13}{2} = 0 \]
To eliminate the fractions, multiply through by 2:
\[ 2x^2 - 11x + 13 = 0 \]

### Final Answer:
The equation formed by increasing each root of \(2x^2 - 3x - 1 = 0\) by 2 is:
\[ 2x^2 - 11x + 13 = 0 \]