Question

find all complex solutions of 4x^2 + 5x + 2 = 0. (if there is more than one solution, separate them with commas.) x =

Explanation:

Step1: Identify coefficients

For the quadratic equation $4x^{2}+5x + 2=0$, we have $a = 4$, $b = 5$, $c = 2$.

Step2: Calculate the discriminant

The discriminant $\Delta=b^{2}-4ac$. Substitute the values: $\Delta = 5^{2}-4\times4\times2=25 - 32=-7$.

Step3: Use the quadratic formula

The quadratic formula is $x=\frac{-b\pm\sqrt{\Delta}}{2a}$. Substitute $a = 4$, $b = 5$, $\Delta=-7$: $x=\frac{-5\pm\sqrt{-7}}{2\times4}=\frac{-5\pm i\sqrt{7}}{8}$.

Answer:

$x=\frac{-5 + i\sqrt{7}}{8},\frac{-5 - i\sqrt{7}}{8}$