Question

solve each equation by using the quadratic formula.
$4x^2 - 4x + 17 = 0$
$\frac{1 + 2i}{2}$
$\frac{1 \pm 4i}{2}$
$\frac{1 + 4i}{2}$
$\frac{4 + 12i\sqrt{2}}{8}$

Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).
Here, \(a = 4\), \(b=-4\), \(c = 17\).

Step2: Calculate the discriminant

Discriminant \(D=b^2-4ac=(-4)^2-4\times4\times17=16 - 272=-256\).

Step3: Substitute into Quadratic Formula

\(x=\frac{-(-4)\pm\sqrt{-256}}{2\times4}=\frac{4\pm16i}{8}\) (since \(\sqrt{-256}=\sqrt{256}\times\sqrt{-1}=16i\)).

Step4: Simplify the fraction

Simplify \(\frac{4\pm16i}{8}=\frac{4(1\pm4i)}{8}=\frac{1\pm4i}{2}\).

Answer:

\(\frac{1\pm4i}{2}\) (corresponding to the option \(\frac{1\pm4i}{2}\))