The number of real solutions of the equation x^4 - 1 = 0 are ________.

Step 1 (Factor the Equation): We want to solve for real numbers $x$ in the equation:

Using the difference of squares identity:

Step 2 (Find all Roots): Setting each factor to zero:

1. $x-1=0⟹x=1$ (Real)

2. $x+1=0⟹x=-1$ (Real)

3. $x^2+1=0⟹x^2=-1⟹x=\pm i$ (Complex / Imaginary)

Step 3 (Count the Real Roots): The real solutions are exactly $x=1$ and $x=-1$. There are exactly 2 real solutions.

Step 4 (Conclusion): The number of real solutions is exactly 2.