Let's analyze the steps in Marika's work and the statements provided.
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Marika's First Step:
- She correctly set up the equation \( (6x + 15)^2 = -24 \). However, a square (like \( (6x + 15)^2 \)) cannot equal a negative number, so it actually indicates that there are no real solutions.
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Step 2 Analysis:
- When Marika attempts to solve \( (6x + 15)^2 = -24 \), she takes the square root of both sides. This is where the problem arises. The square root of a negative number is indeed imaginary, and it should be expressed in terms of \( i \) (the imaginary unit). Specifically, we have \( \sqrt{-24} = \sqrt{24}i = 2\sqrt{6}i \).
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Step 3:
- Marika writes \( 6x + 15 = -2\sqrt{6} \). However, she missed including the \( i \) since she is taking the square root of a negative number. It should be \( 6x + 15 = \pm 2\sqrt{6}i \).
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Step 4:
- In this step, she again misses the imaginary unit \( i \) when she states \( 6x = -2\sqrt{6} - 15 \).
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Step 5:
- The final solution also lacks the imaginary unit.
After this analysis, let's review the statements provided:
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"In step 2, Marika should have simplified \( \sqrt{-24} \) to be \( 4\sqrt{-6} \)." – This is incorrect because she should have expressed it in terms of \( i \), and the proper simplification is \( \sqrt{-24} = 2\sqrt{6}i \).
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"There are no real solutions to this equation because the square root of a negative number is not a real number." – This statement is true. Since \( (6x + 15)^2 = -24 \), there are indeed no real solutions.
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"There should be two real solutions to this equation because \( \sqrt{-24} = \pm 2\sqrt{6} \)." – This statement is false because \( \sqrt{-24} \) is imaginary.
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"Marika correctly solved this equation." – This is false, as she made errors in handling the square root of a negative number.
Based on the analysis, the most accurate statement is:
"There are no real solutions to this equation because the square root of a negative number is not a real number."
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