Assuming an equation with one side as a squared variable expression and the other side is a numeric expression, which statement is correct:

Question

1.the numeric expression is zero, there are two solutions.
2. the numeric expression is zero, there is 1 solution.
3. there are no solutions.

Answer 1

To analyze the situation where one side of the equation is a squared variable expression and the other side is a numeric expression, we can express this in a general form:

\[ x^2 = k \]

where \( k \) is the numeric expression. We'll evaluate the correct statements based on the value of \( k \):

1. If \( k = 0 \):
   The equation becomes \( x^2 = 0 \). The only solution is \( x = 0 \), since the square of any real number is non-negative. Therefore, this scenario corresponds to:

   • Statement 2: the numeric expression is zero, there is 1 solution.

2. If \( k > 0 \):
   The equation becomes \( x^2 = k \). In this case, there are two solutions: \( x = \sqrt{k} \) and \( x = -\sqrt{k} \). So, the statement would be:

   • There are two solutions.

3. If \( k < 0 \):
   The equation becomes \( x^2 = k \). Since a squared expression is always non-negative, there are no real solutions for this case. Thus, the statement would be:

   • Statement 3: there are no solutions.

Summarizing the outcomes:

• If \( k = 0 \): 1 solution.
• If \( k > 0 \): 2 solutions.
• If \( k < 0 \): no solutions.

Therefore, the correct answer when the numeric expression is zero is Statement 2: the numeric expression is zero, there is 1 solution.

Answer 2

But for example, if they give you x^2=16 I would take square root of both sides and get (x-4)=0 and (x+4)=0 So, here I would have 2 answers. Is this related to the question or this is a completely different issue? Thanks

Answer 3

Yes, your example \( x^2 = 16 \) is indeed related to the question, but it falls under a different scenario where the numeric expression is positive.

In your example:

• The equation \( x^2 = 16 \) indicates that the numeric expression (16) is greater than zero.
• When you take the square root of both sides, you correctly form two equations:
  • \( x - 4 = 0 \) which gives you \( x = 4 \)
  • \( x + 4 = 0 \) which gives you \( x = -4 \)

Thus, in this case, there are indeed two solutions: \( x = 4 \) and \( x = -4 \).

To clarify how this connects to the original question about the numeric expression being zero:

• In the original framing, we were discussing three scenarios based on different values for the numeric expression \( k \):
  1. If \( k = 0\): (Here, \( x^2 = 0 \), leading to one solution: \( x = 0 \))
  2. If \( k > 0\): (As in your case, \( x^2 = k \), leading to two solutions)
  3. If \( k < 0\): (No real solutions, because no real number squared will yield a negative result)

So, in summary:

• Your example \( x^2 = 16 \) falls under case 2 (where \( k > 0 \)) and results in two solutions.
• The clarification regarding \( k = 0\) states there is exactly one solution.

Each scenario provides different possibilities based on the value of the numeric expression associated with the squared variable. Thank you for pointing that out!

Answer 4

But with the details and information in the question, how come I should assume that they are talking about a zero, it could be any number? Or not?

Answer 5

You're correct that the original phrasing of the question can lead to some ambiguity, and it is important to clarify that the analysis is based on specific hypothetical cases for the numeric expression.

When the question states "assuming an equation with one side as a squared variable expression and the other side as a numeric expression," it implies that there are various possibilities for the numeric expression. Here’s how to think about it step-by-step:

1. Generic Case: In a general equation of the form \(x^2 = k\), \(k\) can represent any real number—negative, zero, or positive.

2. Different Scenarios:

   • If \(k > 0\) (like in your example with \(x^2 = 16\)), there are two solutions.
   • If \(k = 0\) (like \(x^2 = 0\)), there is one solution.
   • If \(k < 0\) (like \(x^2 = -1\)), there are no real solutions.

3. Assumptions in the Original Question: The original question seems to present a specific condition when it mentions "the numeric expression is zero." When referring to this particular case, the answer must correspond to the situation where \(k = 0\).

4. Context Matters: When discussing the properties of squared expressions, it's often essential to specify what \(k\) represents. If \(k\) were unspecified in a real-world scenario, you would indeed consider all possibilities for \(k\). However, in this structured context where only the case of \(k = 0\) is under scrutiny, the focus narrows specifically on that scenario.

To summarize:

• Yes, \(k\) can technically be any real number in any general problem.
• The original question asks specifically about what happens when the numeric expression is zero, so in that context, the correct answer would be (2): "there is 1 solution," under the assumption that the numeric expression is zero.
• To properly analyze different cases, understanding the nature of \(k\) in various contexts is crucial.

If the question had simply asked about the general situation without focusing on the numeric expression being zero, then you would address all scenarios accordingly. Thank you for bringing up this important nuance!