Question

Which equation has a different solution than the rest?(1 point)
Responses

x4=−1
Start Fraction x over 4 End Fraction equals negative 1

x−2=2
Start Fraction x over negative 2 equals 2

16x=64
16x equals 64

−12x=48

Answer 1

Let's solve each equation to find their solutions:

1. **Equation:** \( x^4 = -1 \)
**Solution:** There are no real solutions because the fourth power of a real number cannot be negative. However, in the complex number system, the solutions would be \( x = e^{i\frac{\pi}{4}} \), \( e^{i\frac{3\pi}{4}} \), \( e^{i\frac{5\pi}{4}} \), and \( e^{i\frac{7\pi}{4}} \).

2. **Equation:** \( \frac{x}{4} = -1 \)
**Solution:** Multiply both sides by 4, \( x = -4 \).

3. **Equation:** \( x - 2 = 2 \)
**Solution:** Solve for \( x \): \( x = 4 \).

4. **Equation:** \( 16x = 64 \)
**Solution:** Divide both sides by 16, \( x = 4 \).

5. **Equation:** \( -12x = 48 \)
**Solution:** Divide both sides by -12, \( x = -4 \).

Now let's summarize the solutions:
1. \( x^4 = -1 \) → no real solution (complex solutions)
2. \( \frac{x}{4} = -1 \) → \( x = -4 \)
3. \( x - 2 = 2 \) → \( x = 4 \)
4. \( 16x = 64 \) → \( x = 4 \)
5. \( -12x = 48 \) → \( x = -4 \)

From this, we see that:
- The equation \( x^4 = -1 \) has complex solutions (no real solution).
- The equations \( \frac{x}{4} = -1 \) and \( -12x = 48 \) have the solution \( x = -4 \).
- The equations \( x - 2 = 2 \) and \( 16x = 64 \) have the solution \( x = 4 \).

Thus, the equation that has a different solution than the rest is:

**\( x^4 = -1 \)**.