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 is zero, there are no solutions.
(1 point)
Statement #
is correct.
Statement #2: If the numeric expression is zero, there is one solution is correct.
The correct statement is Statement #2: If the numeric expression is zero, there is one solution.
When the numeric expression on one side of the equation is zero, it means that the square variable expression on the other side must also be zero in order for the equation to be true. This results in one solution where the variable takes on a value that makes the equation true.
To determine which statement is correct, we need to analyze the equation. In this case, we have an equation with one side as a squared variable expression and the other side as a numeric expression. Let's say the equation is represented as follows:
ax^2 + bx + c = 0,
where a, b, and c are constants, and x is the variable we are trying to solve for.
Statement #1: If the numeric expression is zero, there are two solutions.
If the numeric expression on the right side of the equation is zero, it means the equation becomes:
ax^2 + bx + c = 0.
In this scenario, if the quadratic equation can be factored or solved using the quadratic formula, it is possible to get two real solutions. This is because a quadratic equation can have a maximum of two real solutions. Therefore, Statement #1 could be correct, but we need further analysis to determine if it is always the case.
Statement #2: If the numeric expression is zero, there is one solution.
If the numeric expression on the right side of the equation is zero, it means the equation becomes:
ax^2 + bx + c = 0.
In this case, if the quadratic equation can be factored into the form (x - p)^2 = 0, it would indicate that there is only one repeated solution (also known as a double root). However, this is not always the case. If the quadratic equation cannot be factored in this form, it may still have two distinct real solutions or complex solutions. Therefore, Statement #2 is not correct.
Statement #3: If the numeric expression is zero, there are no solutions.
If the numeric expression on the right side of the equation is zero, it means the equation becomes:
ax^2 + bx + c = 0.
In this situation, if the quadratic equation cannot be factored or solved using the quadratic formula, it implies that there are no real solutions. The equation may have two complex solutions, but no real solutions. Hence, Statement #3 is not correct.
Based on the analysis, the correct statement is Statement #1: If the numeric expression is zero, there are two solutions. However, it is important to note that this is not always the case. It depends on whether the quadratic equation can be factored or solved using the quadratic formula.