To provide a detailed explanation of the errors made by an incorrect student, I need more context about the specific models and equations being referenced. However, I can illustrate a general approach to analyzing student errors using mathematical models and equations.
Let's assume the student is working on a problem involving a quadratic equation, governed by the standard form \( ax^2 + bx + c = 0 \), where the student needs to apply the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here are some common types of errors a student might make, along with explanations referencing the model and equations:
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Misapplication of the quadratic formula:
- Error: The student may incorrectly identify \( a, b, \) and \( c \) from the given quadratic equation.
- Example: For the equation \( 3x^2 + 12x + 9 = 0 \), the student might incorrectly use \( a = 3 \), \( b = -12\) instead of \( b = 12 \), and \( c = 9 \).
- Explanation: Substituting incorrect coefficients into the quadratic formula will yield incorrect values for \( x \) because the structure of the formula directly relies on correct values for \( a, b, \) and \( c \).
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Calculation Errors:
- Error: The student might miscalculate the discriminant \( b^2 - 4ac \).
- Example: For the correct coefficients \( a = 3 \), \( b = 12 \), and \( c = 9\), the discriminant should be calculated as \( 12^2 - 4 \cdot 3 \cdot 9\).
- Explanation: If the student calculates \( 12^2 = 144\) and \( 4 \cdot 3 \cdot 9 = 108\), they should get \( 144 - 108 = 36\) (which is correct). However, if they mistakenly compute \( 144 - 120 \), they will think the discriminant is 24, leading them to an erroneous conclusion about the nature of the roots (since a positive discriminant indicates two distinct real roots, while a negative discriminant indicates no real solutions).
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Improper simplification:
- Error: The student may improperly simplify expressions when computing the answers.
- Example: If simplifying \( -12 \pm \sqrt{36} \), the student could mistakenly think \( \sqrt{36} = 6 \) results in \( -12 \pm 6 \) yielding solutions of \( -3 \) and \( -18 \) instead of the correct \( -6 \) and \( -18 \).
- Explanation: Simplifying expressions incorrectly can lead to inaccurate solutions. Correct simplification leads to \( x = \frac{-12 + 6}{6} = -1\) and \( x = \frac{-12 - 6}{6} = -3\).
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Sign Errors:
- Error: The student may overlook a negative sign while solving.
- Example: If the calculation leads to \( x = \frac{-b}{2a} \), confusion can arise over whether \( b \) should be negated.
- Explanation: The negative sign is crucial as it changes the value of \( x \). Not applying it correctly leads to incorrect solutions for the roots of the equation.
To summarize, students may make errors due to misidentification of parameters, arithmetic mistakes, improper simplification, or sign errors. Each of these errors can be traced back to misapplication of the rules of algebra and improper handling of the quadratic formula, with consequences that cascade through to the final answers.
For a more specific analysis, I would need the exact problem the student was working on and the erroneous steps taken.
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