Solve the Equation: x⁴ + x² = 0 Using Factoring

The problem at hand is to solve the equation $x^4 + x^2 = 0$.

Let's begin by factoring the expression:

The given equation is: $x^4 + x^2 = 0$

We can factor out the common factor of $x^2$ from both terms:

$x^2(x^2 + 1) = 0$

To solve for $x$, we set each factor equal to zero:

• $x^2 = 0$

Solving for $x$, we have:

$x = 0$

Next, consider the second factor:

• $x^2 + 1 = 0$

Solving for $x$, we have:

$x^2 = -1$

Since $x^2 = -1$ has no real solutions, we ignore these solutions in the real number system.

Thus, the only real solution to the equation $x^4 + x^2 = 0$ is:

$x = 0$