To solve the quadratic equation \( 14x^2 + 2x + 4 = 0 \), we can use the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]
where \( a = 14 \), \( b = 2 \), and \( c = 4 \).
First, we will calculate the discriminant (\( b^2 - 4ac \)):
\[ b^2 = (2)^2 = 4 \] \[ 4ac = 4 \cdot 14 \cdot 4 = 224 \] \[ b^2 - 4ac = 4 - 224 = -220 \]
Since the discriminant is negative (\(-220\)), this means that the quadratic equation has no real solutions but has two complex solutions.
Now, we can calculate the solutions using the quadratic formula:
\[ x = \frac{{-2 \pm \sqrt{-220}}}{2 \cdot 14} \]
First, calculate \( \sqrt{-220} \):
\[ \sqrt{-220} = \sqrt{220}i = \sqrt{4 \cdot 55}i = 2\sqrt{55}i \]
Now substitute this back into the quadratic formula:
\[ x = \frac{{-2 \pm 2\sqrt{55}i}}{28} \]
Simplifying:
\[ x = \frac{{-1 \pm \sqrt{55}i}}{14} \]
Thus, the solutions to the equation \( 14x^2 + 2x + 4 = 0 \) are:
\[ x = \frac{{-1 + \sqrt{55}i}}{14} \quad \text{and} \quad x = \frac{{-1 - \sqrt{55}i}}{14} \]