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.
Statement #2: If the numeric expression is zero, there is one solution.
Statement #1: If the numeric expression is zero, there are two solutions.
This statement is correct. If the numeric expression on one side of the equation is zero, and the other side is a squared variable expression, then there are usually two solutions. This is because the squared expression can be both positive and negative, leading to two possible values that satisfy the equation.
To determine which statement is correct, we need to consider the nature of a squared variable expression in an equation.
If we have an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, then we can determine the number of solutions based on the discriminant, which is the term inside the square root of the quadratic formula.
The discriminant is given by the formula D = b^2 - 4ac.
Based on this information, we can evaluate each statement:
Statement #1: If the numeric expression is zero, there are two solutions.
This statement is correct. If the numeric expression is zero, it means that the discriminant D is positive. In this case, the equation has two distinct real solutions.
Statement #2: If the numeric expression is zero, there is one solution.
This statement is incorrect. If the numeric expression is zero, it means that the discriminant D is zero. In this case, the equation has one real solution, known as a repeated root.
Statement #3: If the numeric expression is zero, there are no solutions.
This statement is incorrect. If the numeric expression is zero, it means that the discriminant D is negative. In this case, the equation does not have any real solutions. Instead, it has two complex conjugate solutions.
Therefore, the correct statement is Statement #1: If the numeric expression is zero, there are two solutions.