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:
\[ x^2 = k \]
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If \( k = 0 \), the equation becomes: \[ x^2 = 0 \] The only solution is: \[ x = 0 \] Therefore, in this case, there is one solution.
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If \( k > 0 \), the equation becomes: \[ x^2 = k \] Here, there are two solutions: \[ x = \sqrt{k} \quad \text{and} \quad x = -\sqrt{k} \]
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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.
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