Question

The quadratic equation, whose roots are \( \frac{4 + \sqrt{7}}{2} \) and \( \frac{4 - \sqrt{7}}{2} \), is:

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

Hint:

For a quadratic equation with known roots \( r_1 \) and \( r_2 \), use the relations: \text{Sum of roots} = -\frac{\text{coefficient of } x}{\text{leading coefficient}}, \quad \text{Product of roots} = \frac{\text{constant term}}{\text{leading coefficient}}.

Answer 1

The Correct Option is C

We are given the roots of the quadratic equation as \( \frac{4 + \sqrt{7}}{2} \) and \( \frac{4 - \sqrt{7}}{2} \). The sum of the roots is: \[ \text{Sum of roots} = \frac{4 + \sqrt{7}}{2} + \frac{4 - \sqrt{7}}{2} = 4 \] The product of the roots is: \[ \text{Product of roots} = \left( \frac{4 + \sqrt{7}}{2} \right) \times \left( \frac{4 - \sqrt{7}}{2} \right) = \frac{16 - 7}{4} = \frac{9}{4} \] The quadratic equation is given by: \[ x^2 - (\text{Sum of roots})x + \text{Product of roots} = 0 \] Substituting the sum and product of the roots: \[ x^2 - 4x + \frac{9}{4} = 0 \] Multiplying through by 4 to clear the fraction: \[ 4x^2 - 16x + 9 = 0 \] Thus, the correct answer is \( 4x^2 - 16x + 9 = 0 \).