Question

assuming an equation with one side as a squared variable expression and the other side as a numeric expression which of the following statements is correct?
statement #1 if the numeric expression is zero there are two solutions.
statement #2 if the numeric expression is zero there are one solution
statement #3 if the numeric expression is zero there are no solutions

Answer 1

To understand the behavior of a squared variable expression set equal to a numeric expression, let's denote the squared variable expression as \( x^2 \) and the numeric expression as \( k \).

When we set up the equation:

\[
x^2 = k
\]

1. **If \( k = 0 \)**, the equation becomes:
\[
x^2 = 0
\]
The only solution is:
\[
x = 0
\]
Therefore, in this case, there is **one solution**.

2. **If \( k > 0 \)**, the equation becomes:
\[
x^2 = k
\]
Here, there are two solutions:
\[
x = \sqrt{k} \quad \text{and} \quad x = -\sqrt{k}
\]

3. **If \( k < 0 \)**, the equation becomes:
\[
x^2 = k
\]
In this case, there are no real solutions since a squared term (like \( x^2 \)) cannot equal a negative number.

Given these analyses, the correct interpretation for when the numeric expression is zero (as per your statements) is:

- **Statement #1:** "if the numeric expression is zero there are two solutions." - **Incorrect**
- **Statement #2:** "if the numeric expression is zero there are one solution." - **Correct**
- **Statement #3:** "if the numeric expression is zero there are no solutions." - **Incorrect**

Therefore, the correct statement is **Statement #2**: if the numeric expression is zero there is one solution.