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?

Question

Statement #1: if the numeric expression is zero, there are two solutions.

Statement #2: if the numeric expression is zero, there is one solution

Statement #3: If the numeric expression is zero, there are no solutions.

(1 point)

Answer 1

The correct statement is:

Statement #2: if the numeric expression is zero, there is one solution.

This is based on the structure of the equation. If we have an equation of the form \( x^2 = 0 \), the only solution is \( x = 0 \). In general, when a squared variable expression equals zero, it results in a single solution, which is the value that makes the expression equal to zero. Thus, if the numeric expression is zero, there is indeed one solution.

Answer 2

are you sure?

Answer 3

Yes, I am sure.

To clarify further, let's consider an equation of the form:

\[ x^2 = k \]

where \( k \) is the numeric expression.

1. If \( k > 0 \): There are two solutions (e.g., \( x^2 = 4 \) gives \( x = 2 \) and \( x = -2 \)).
2. If \( k = 0 \): There is one solution (e.g., \( x^2 = 0 \) gives \( x = 0 \)).
3. If \( k < 0 \): There are no real solutions (e.g., \( x^2 = -4 \) has no real solutions).

So specifically for the case when the numeric expression is zero (\( k = 0 \)), there is indeed only one solution, which is \( x = 0 \).

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