Question

Solve for x .
x4 − 2 = −10

Answer 1

To solve the equation \( x^4 - 2 = -10 \), we start by simplifying it.

1. Add 2 to both sides of the equation:

\[
x^4 - 2 + 2 = -10 + 2
\]

This simplifies to:

\[
x^4 = -8
\]

2. Now, we need to solve for \( x \). However, since \( x^4 \) cannot equal a negative number (as real numbers raised to an even power must be non-negative), we conclude that there are no real solutions.

3. If we consider complex solutions, we can rewrite the equation as:

\[
x^4 = -8
\]

4. To express this in terms of complex numbers, we can write \(-8\) in exponential form. We can interpret \(-8\) in polar coordinates:

\[
-8 = 8(\cos(\pi) + i\sin(\pi)) = 8e^{i\pi}
\]

5. We are looking for the fourth roots of \(-8\). The formula for finding the \( n \)-th roots of a complex number \( re^{i\theta} \) is given by:

\[
z_k = r^{1/n} e^{i(\theta + 2k\pi)/n}
\]

for \( k = 0, 1, 2, \ldots, n-1 \).

Here, \( r = 8 \), \( \theta = \pi \), and \( n = 4 \).

6. Therefore, the magnitude of the roots is:

\[
r^{1/4} = 8^{1/4} = 2
\]

7. The angles for the fourth roots are:

\[
\theta_k = \frac{\pi + 2k\pi}{4} = \frac{\pi(1 + 2k)}{4} \quad \text{for } k = 0, 1, 2, 3
\]

So the angles for the values of \( k \) are:

- For \( k = 0 \): \( \theta_0 = \frac{\pi}{4} \)
- For \( k = 1 \): \( \theta_1 = \frac{3\pi}{4} \)
- For \( k = 2 \): \( \theta_2 = \frac{5\pi}{4} \)
- For \( k = 3 \): \( \theta_3 = \frac{7\pi}{4} \)

8. Thus, the four complex solutions are:

\[
x_0 = 2e^{i\pi/4} = 2\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) = 2\left(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right) = \sqrt{2} + i\sqrt{2}
\]

\[
x_1 = 2e^{i3\pi/4} = 2\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right) = 2\left(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right) = -\sqrt{2} + i\sqrt{2}
\]

\[
x_2 = 2e^{i5\pi/4} = 2\left(\cos\frac{5\pi}{4} + i\sin\frac{5\pi}{4}\right) = 2\left(-\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\right) = -\sqrt{2} - i\sqrt{2}
\]

\[
x_3 = 2e^{i7\pi/4} = 2\left(\cos\frac{7\pi}{4} + i\sin\frac{7\pi}{4}\right) = 2\left(\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}\right) = \sqrt{2} - i\sqrt{2}
\]

Therefore, the four complex solutions to the equation \( x^4 - 2 = -10 \) are:

\[
x = \sqrt{2} + i\sqrt{2}, \ -\sqrt{2} + i\sqrt{2}, \ -\sqrt{2} - i\sqrt{2}, \ \text{and} \ \sqrt{2} - i\sqrt{2}
\]