To evaluate the given statements about an equation of the form \( ax^2 + bx + c = k \) (where \( ax^2 + bx + c \) is a squared variable expression and \( k \) is a numeric expression), we can rearrange it to the standard form:
\[ ax^2 + bx + (c - k) = 0 \]
We need to analyze the situation when \( c - k > 0 \):
- Statement #1: If the numeric expression is greater than zero, there are two solutions.
- Statement #2: If the numeric expression is greater than zero, there is one solution.
- Statement #3: If the numeric expression is greater than zero, there are no solutions.
The number of solutions to a quadratic equation can be determined using the discriminant, \( D = b^2 - 4ac \):
- If \( D > 0 \), there are two distinct real solutions.
- If \( D = 0 \), there is exactly one real solution (a repeated root).
- If \( D < 0 \), there are no real solutions (the solutions are complex).
Given our context, if the numeric expression (let's denote it \( k \)) is greater than zero, we must consider the entire expression:
- If \( c - k > 0 \), since \( c \) itself must be greater than \( k \), we may have positive \( a \) or \( b \) depending on their values. Therefore, depending on the discriminant \( D \), it can lead to 2 solutions, 1 solution or even none (but this scenario generally arises due to particular values of \( b \) and \( a \)).
- If \( k \) stays greater than \( c \), eventually reducing \( (c - k) \leq 0 \) can cause no solutions associated with the quadratic characteristic.
Now simplified:
- Therefore, Statement #1 is generally true when considering cases leading to the positive discriminant, yielding two solutions more frequently as quadratics are quadratic functions up until the vertex regardless of constant placement others.
Thus, the correct choice from the provided statements based on discussion should be:
Statement #1 is correct.
3 answers