Question

\sqrt{x} = -3

Explanation:

🆕 New Concept Discovered: Solving Radical Equations
The principal square root of a real number is always non-negative.

Step 1: Analyze the equation

We are given the equation:

$$ \sqrt{x} = -3 $$

By definition, the principal square root symbol \( \sqrt{\cdot} \) represents the non-negative square root of a number. This means that for any real number \( x \geq 0 \):

$$ \sqrt{x} \geq 0 $$

Step 2: Evaluate for real solutions

Since the left side of the equation \( \sqrt{x} \) must be greater than or equal to \( 0 \), and the right side is a negative number (\(-3\)), there is no real number \( x \) that can satisfy this equation.

If we attempt to solve by squaring both sides:

$$ (\sqrt{x})^2 = (-3)^2 $$

$$ x = 9 $$

We must check this extraneous solution in the original equation:

$$ \sqrt{9} = 3 eq -3 $$

Thus, \( x = 9 \) is an extraneous solution and not a valid solution.

Answer:

No solution (or no real solution)