Question

Solve the following quadratic equation: 14x2+2x+4=0

Answer 1

To solve the quadratic equation \(14x^2 + 2x + 4 = 0\), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, the coefficients are:
- \(a = 14\)
- \(b = 2\)
- \(c = 4\)

First, we calculate the discriminant (\(D\)):

\[
D = b^2 - 4ac
\]
\[
D = 2^2 - 4 \cdot 14 \cdot 4
\]
\[
D = 4 - 224
\]
\[
D = -220
\]

Since the discriminant is negative, this means that the quadratic equation has no real solutions and instead has two complex solutions.

Now we can find the complex solutions using the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{-220}}{2 \cdot 14}
\]

We simplify \(\sqrt{-220}\):

\[
\sqrt{-220} = \sqrt{220} i = \sqrt{4 \cdot 55} i = 2\sqrt{55} i
\]

Now substitute this back into the formula:

\[
x = \frac{-2 \pm 2\sqrt{55} i}{28}
\]
\[
x = \frac{-1 \pm \sqrt{55} i}{14}
\]

Thus, the solutions to the equation \(14x^2 + 2x + 4 = 0\) are:

\[
x = \frac{-1 + \sqrt{55} i}{14}, \quad x = \frac{-1 - \sqrt{55} i}{14}
\]