To solve the equation:

$\sqrt{x^2} = x$

we need to understand the behavior of the square root function. The square root of $x^2$ can be simplified as follows:

$\sqrt{x^2} = |x|$

So, the equation becomes:

$|x| = x$

The absolute value $|x|$ is defined as:

• $x$ when $x \geq 0$
• $-x$ when $x < 0$

For $|x| = x$ to hold true, $x$ must be non-negative, as otherwise $|x|$ would not equal $x$. Therefore, the solution is:

$x \geq 0$

In interval notation, the solution is:

$x \in [0, \infty)$

Would you like further clarification on this solution?

Here are five related questions for further exploration:

1. What if the equation were $\sqrt{x^2} = -x$? How would the solution change?
2. How would the solution differ if the equation were $\sqrt{x} = x$?
3. What are the properties of the absolute value function, and how does it impact solutions to equations?
4. Can we have a similar scenario with a cube root, like $\sqrt[3]{x^2} = x$?
5. How does the concept of absolute value affect inequalities involving square roots?

Tip: When dealing with square roots, always remember that the principal square root (i.e., $\sqrt{x^2}$) represents the non-negative value, which can lead to absolute values in equations.