To determine the number of solutions for 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 analyze the situation depending on the value of k.
Assuming we rewrite the equation as \( ax^2 + bx + (c-k) = 0 \) (i.e., moving all terms to one side), we will look at the discriminant (\( D \)) given by:
\[ D = b^2 - 4a(c-k) \]
The number of solutions depends on the sign of the discriminant:
- 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.
Now, focusing on your scenario where \( k \) (the numeric expression) is greater than zero, we can analyze the implications:
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If \( c - k < 0 \): This means \( c < k \). Thus, \( D = b^2 - 4a(c-k) \) will depend on the value of \( b^2 \) compared to \( -4a(c-k) \). Depending on the values of \( a, b, \text{ and } c\), it’s possible to have two, one, or no solutions.
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If \( c - k = 0 \): Then \( D = b^2 \). If \( b^2 \) is greater than or equal to zero, there will be one solution.
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If \( c - k > 0 \): This means \( c > k \). In this case, it generally suggests there may be two solutions if \( D > 0 \), but again depends on specific values.
Thus, merely stating that "the numeric expression (k) is greater than zero" does not definitively lead to one of the three scenarios (two solutions, one solution, or no solutions).
So, the correct conclusion with regard to the general case is:
- If the numeric expression (k) is greater than zero, there is not a definitive conclusion about whether there are two, one, or no solutions without further information regarding the coefficients \( a \), \( b \), and \( c \).
Therefore, the most accurate statement in general terms regarding quadratic expressions and their solutions is: Not enough information to determine if there are one, two, or no solutions solely based on k being greater than zero.
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