To analyze the statements regarding an equation of the form \( ax^2 + bx + c = 0 \) (where one side is a squared variable expression and the other side is a numeric expression), we need to consider the scenario when the numeric expression is zero. This leads us to consider the equation in the standard form:
\[ ax^2 + bx + c = 0. \]
When we set the numeric expression to zero, the equation simplifies to:
\[ ax^2 + bx + c = 0. \]
Now, we will evaluate each statement based on the nature of the quadratic equation:
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Statement 1: If the numeric expression is zero, there are two solutions.
- This statement is not always correct. A quadratic equation can have two, one, or no real solutions depending on the discriminant \( D = b^2 - 4ac \). If \( D > 0 \), there are two distinct solutions. If \( D = 0 \), there is exactly one solution (a repeated root). If \( D < 0 \), there are no real solutions.
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Statement 2: If the numeric expression is zero, there is one solution.
- This statement is not always correct as well. There is one solution only when the discriminant \( D = 0 \).
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Statement 3: If the numeric expression is zero, there are no solutions.
- This statement is not always correct. A quadratic equation can have no solutions when \( D < 0 \), but it can also have one or two solutions depending on the coefficients of the equation.
In summary, none of the statements can be deemed correct universally because the number of solutions depends on the specific values of the coefficients in the quadratic equation. Specifically, if the numeric expression (right side) is zero, the number of solutions is determined by the discriminant \( D \):
- Two solutions if \( D > 0 \),
- One solution if \( D = 0 \),
- No solutions if \( D < 0 \).
Thus, the correct assertion regarding the nature of solutions when the numeric expression is zero is contingent upon the discriminant, and not any of the statements alone can be accepted as correct without that context.
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