Question

what are the solutions to the quadratic equation $4x^2 - 4x + 5 = 0$? choose the correct answer from the choices. $\bigcirc$ $x = 2 + i$ and $x = 2 - i$ $\bigcirc$ $x = 1 + 2i$ and $x = 1 - 2i$ $\bigcirc$ $x = -1 + \frac{5}{2}i$ and $x = -1 - \frac{5}{2}i$ $\bigcirc$ $x = \frac{1}{2} + i$ and $x = \frac{1}{2} - i$

Explanation:

Step1: Recall quadratic formula

For $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Step2: Identify a, b, c

$a=4$, $b=-4$, $c=5$

Step3: Calculate discriminant

$\Delta = (-4)^2-4(4)(5)=16-80=-64$

Step4: Compute square root of discriminant

$\sqrt{\Delta}=\sqrt{-64}=8i$

Step5: Substitute into quadratic formula

$x=\frac{4\pm8i}{2\times4}=\frac{4\pm8i}{8}$

Step6: Simplify the expression

$x=\frac{4}{8}\pm\frac{8i}{8}=\frac{1}{2}\pm i$

Answer:

$\boldsymbol{x=\frac{1}{2}+i}$ and $\boldsymbol{x=\frac{1}{2}-i}$ (the last option)