To solve the equation \( x^4 - 2 = -10 \), we start by simplifying it.
- Add 2 to both sides of the equation:
\[ x^4 - 2 + 2 = -10 + 2 \]
This simplifies to:
\[ x^4 = -8 \]
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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.
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If we consider complex solutions, we can rewrite the equation as:
\[ x^4 = -8 \]
- To express this in terms of complex numbers, we can write \(-8\) in exponential form. We can interpret \(-8\) in polar coordinates:
\[ -8 = 8(\cos(\pi) + i\sin(\pi)) = 8e^{i\pi} \]
- 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 \( n = 4 \).
- Therefore, the magnitude of the roots is:
\[ r^{1/4} = 8^{1/4} = 2 \]
- 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 \( k = 3 \): \( \theta_3 = \frac{7\pi}{4} \)
- 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:
\[ x = \sqrt{2} + i\sqrt{2}, \ -\sqrt{2} + i\sqrt{2}, \ -\sqrt{2} - i\sqrt{2}, \ \text{and} \ \sqrt{2} - i\sqrt{2} \]
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