To determine whether the student's work on an equation is correct or incorrect, we need to know the specific equation they solved and see how they approached the solution. Here are some guidelines based on how a linear or polynomial equation can yield different numbers of solutions:
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Infinitely Many Solutions: This occurs when the equation simplifies to an identity, such as \(0 = 0\). For example, if after simplifying, you end up with a true statement regardless of the variable's value (like \(x - x = 0\)).
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One Solution: This occurs when the equation can be rearranged to find a unique value for the variable. For instance, \(2x + 4 = 10\) can be solved to yield \(x = 3\).
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Two Solutions: This is typical for quadratic equations set to zero (i.e., \(ax^2 + bx + c = 0\)), where the discriminant is positive. For example, \(x^2 - 3x + 2 = 0\) factors into \((x - 1)(x - 2) = 0\), giving solutions \(x = 1\) and \(x = 2\).
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Three Solutions: For polynomial equations of degree three, it is possible to have three distinct real solutions (e.g., \(x^3 - 6x^2 + 11x - 6 = 0\) factors to \((x - 1)(x - 2)(x - 3) = 0\), yielding \(x = 1, 2, 3\)).
Without the specific equation or the student's work details, I can't definitively say if the solutions are correct, but these are the general outcomes based on the nature of the equation. If you can provide me with the equation or steps taken by the student, I can analyze it further to determine if the solution is correct and identify the possible number of solutions.
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