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

Statement #____ is correct.

Answer 1

To evaluate the statements regarding an equation with one side as a squared variable expression and the other side as a numeric expression, let's consider a general equation of the form:

\[ ax^2 = c \]

where \( a \) is a non-zero constant, \( x \) is the variable, and \( c \) is the numeric expression.

Now, if \( c = 0 \), the equation becomes:

\[ ax^2 = 0 \]

To find the solutions, we can solve for \( x \):

1. Divide both sides by \( a \) (since \( a \neq 0 \)):
\[ x^2 = 0 \]

2. Taking the square root of both sides gives:
\[ x = 0 \]

In this case, there is exactly **one solution**, which is \( x = 0 \).

Now let's evaluate the statements:

- **Statement #1:** If the numeric expression is zero, there are two solutions. (Incorrect)
- **Statement #2:** If the numeric expression is zero, there is one solution. (Correct)
- **Statement #3:** If the numeric expression is zero, there are no solutions. (Incorrect)

Thus, the correct statement is:

Statement #**2** is correct.