Question

Assuming an equation with one side as a squared variable expression and the other side as numeric expression, which of the following statements is correct?

Answer 1

To determine which statement is correct regarding an equation with one side as a squared variable expression and the other side as a numeric expression, we need to analyze the general form of such equations.

An example of such an equation is:

\[
x^2 = k
\]

where \(x\) is a variable and \(k\) is a numerical expression (a constant).

Based on this format, we can make the following observations:

1. **Solutions**: The solutions will depend on the value of \(k\):
- If \(k > 0\), there are two real solutions: \(x = \sqrt{k}\) and \(x = -\sqrt{k}\).
- If \(k = 0\), there is one real solution: \(x = 0\).
- If \(k < 0\), there are no real solutions (the solutions will be complex).

2. **Nature of the Squared Term**: The squared term \(x^2\) is always non-negative for all real \(x\). This means that if you have an equation where the squared term on one side is equal to a negative number on the other side, then there are no real solutions.

3. **Graphical Interpretation**: The graph of \(y = x^2\) is a parabola opening upwards. If you graph a horizontal line \(y = k\), the number of intersection points (solutions) between the line and the parabola will depend on the value of \(k\).

4. **Manipulation and Restrictions**: When solving for \(x\), we often take a square root, which introduces the necessity to consider both the positive and negative roots when \(k > 0\). This can lead to extraneous solutions if one doesn't carefully consider the context of the original equation.

If you provide specific statements or options, I could help determine which one is correct based on the above analysis.