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

1 answer

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.