The quadratic equation ,whose roots are `(4+sqrt(7))/(2)and(4-sqrt(7))/(2)` is

Options

A. `4x^(2)+16x+9=0`
B. `4x^(2)-16x-9=0`
C. `4x^(2)-16x+9=0`
D. `4x^(2)+16x-9=0`

Answer

To find the quadratic equation whose roots are \(\frac{4+\sqrt{7}}{2}\) and \(\frac{4-\sqrt{7}}{2}\), we can use the relationship between the roots and coefficients of a quadratic equation. The general form of a quadratic equation is given by:

\[ x^2 - (sum \ of \ roots)x + (product \ of \ roots) = 0 \]

### Step 1: Calculate the Sum of the Roots
The roots are:
\[ r_1 = \frac{4+\sqrt{7}}{2}, \quad r_2 = \frac{4-\sqrt{7}}{2} \]
The sum of the roots \(S\) is:
\[ S = r_1 + r_2 = \frac{4+\sqrt{7}}{2} + \frac{4-\sqrt{7}}{2} \]
Combining the fractions:
\[ S = \frac{(4+\sqrt{7}) + (4-\sqrt{7})}{2} = \frac{4 + 4}{2} = \frac{8}{2} = 4 \]

### Step 2: Calculate the Product of the Roots
The product of the roots \(P\) is:
\[ P = r_1 \cdot r_2 = \left(\frac{4+\sqrt{7}}{2}\right) \cdot \left(\frac{4-\sqrt{7}}{2}\right) \]
Using the difference of squares:
\[ P = \frac{(4+\sqrt{7})(4-\sqrt{7})}{4} = \frac{4^2 - (\sqrt{7})^2}{4} = \frac{16 - 7}{4} = \frac{9}{4} \]

### Step 3: Form the Quadratic Equation
Now that we have the sum and product of the roots, we can substitute these into the quadratic equation format:
\[ x^2 - (sum \ of \ roots)x + (product \ of \ roots) = 0 \]
Substituting \(S = 4\) and \(P = \frac{9}{4}\):
\[ x^2 - 4x + \frac{9}{4} = 0 \]

### Step 4: Eliminate the Fraction
To eliminate the fraction, multiply the entire equation by 4:
\[ 4x^2 - 16x + 9 = 0 \]

### Conclusion
Thus, the quadratic equation whose roots are \(\frac{4+\sqrt{7}}{2}\) and \(\frac{4-\sqrt{7}}{2}\) is:
\[ \boxed{4x^2 - 16x + 9 = 0} \]