Question

Find the roots of the quadratic equation \( 4x^2 + 3x + 5 = 0 \).

Hint:

For a negative discriminant, the roots are complex, and you can express them as a complex number \( x = \frac{-b \pm i\sqrt{D}}{2a} \), where \( D \) is the discriminant.

Answer 1

We are given the quadratic equation: \[ 4x^2 + 3x + 5 = 0. \] To find the roots of the quadratic equation, we will use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 4 \), \( b = 3 \), and \( c = 5 \). Substitute the values of \( a \), \( b \), and \( c \) into the formula: \[ x = \frac{-3 \pm \sqrt{3^2 - 4(4)(5)}}{2(4)} = \frac{-3 \pm \sqrt{9 - 80}}{8} = \frac{-3 \pm \sqrt{-71}}{8}. \] Since the discriminant \( \sqrt{-71} \) is negative, the roots are complex. \[ x = \frac{-3 \pm i\sqrt{71}}{8}. \] Thus, the roots of the quadratic equation are: \[ x_1 = \frac{-3 + i\sqrt{71}}{8}, \quad x_2 = \frac{-3 - i\sqrt{71}}{8}. \]