Morose
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Trainerblue192
Charlotte
Lydia
Charlotte made her way to the library. She anticipated meeting Lydia in the library. Charlotte was already a little late. She had gotten distracted with the scene in the garden. She did not like being late, but there was nothing to fix that now. Charlotte hurried, arriving in the library a little short of breath.
The library was large, tastefully decorated, and a beautiful chandelier. There were small areas for studying, including by a fireplace. One of Charlotte’s favorite places during the colder months.
Loath to move as Lydia was, having her as a tutor meant meeting her at her habitual spot. Tucked away in a distant corner, it was, despite the nearby window, a bit dim for how the window never caught direct sunlight. It straddled a fine line between dreary and cozy. At Charlotte's arrival, Lydia’s mice scattered, and she promptly set aside her writings.
”Miss Wrottesley, forgive my tardiness,” she said, claiming a seat. She did not explain why she was late. ”I hope you have had a pleasant morning.”
Lydia nodded along as she collected her spread of texts, saved her place, and set them aside.
“You’ve never heard me complain about more time to read, have you?” she stated. She produced fresh pages from her notes and set them and her quill and ink on the table. “Now then, what are we reviewing today?”
Charlotte smiled, ”That is true.” Charlotte set her notes on the table, ”It would seem that we are doing—” Charlotte showed Lydia her notes.
“These would be cubic equations,” interrupted Lydia. She pointed with her pinkie at one of the equations Charlotte had written down. A little smile grew in the corner of Lydia’s mouth. “New material, excellent. How are you finding it? Specific points of trouble, or shall we go over the matter in full?”
Charlotte’s heart was racing, she felt trapped suddenly, like she needed to stand and pace around the room. She glanced out the window trying to compose herself. ”I do not understand how there are three or one answers. I am uncertain how to factorize. I...” She trails off. ”I am lost.”
Lydia nodded knowingly. She reviewed the notes in silence, occasionally nodding. In short order, she pushed the notes back and set blank paper in front of Charlotte.
“Let’s simplify all of this and start from the top, then.” She took her quill and wrote out an equation, reading it aloud as she wrote. “Two-X cubed, plus three-X squared, plus two-X, plus three equals zero. This is a basic cubic equation. Actually, let’s cross this out. Two-X cubed, plus three-X squared, plus four-X, plus five equals zero. These are both cubic equations, but this second one will make it easier for you. Now, a basic cubic equation follows the form of A-X cubed, plus B-X squared, plus C-X, plus D equals zero. So taking the equation from before, let’s compare. A is two and B is three. Do you follow? What is C, and what is D?”
Charlotte froze. How could anything with that many pluses equal zero? ”C is four and D is five?” Charlotte said uncertainty. It sounded fake. But also how did all that still equal zero? ”But, how? Does the cubing not make it bigger? I know to square something is part of the number. But if you are adding numbers together how do they equal zero?”
Lydia sighed and shook her head. “We’re drawing connections between a template and an example. Each letter is what we call a variable, which is a number that we don’t know the value of, but which remains consistent. Each number written to the left is multiplying that given variable by however many. So when I say Two-X cubed, what I am saying is that there is a number, X, which is being multiplied by itself three times—that being the meaning of the term cubed, and then that X-cubed is multiplied by two. X is the same each time it appears in the equation; what it’s multiplied by is what differs. Why is the result zero? That’s for us to find out using mathematical processes. But there should be at least some number, represented by X, that when put into this equation in the place of X, will result in this equation equalling zero. We see these other letters, A, B, C, and D in the cubic archetype because they’re stand-ins.
Now, with that said, let’s return to the task at hand. The way to solve these problems is by using Cardano’s Formula. According to Cardano, a cubic of the form one sees here will result in three potential values for x. I see your notes have the Proof for it. I ask you again, as I have many times. Do you feel it necessary to understand the Proof, or will the ability to use it suffice?”
”I would like to understand it. For now the ability to use it shall suffice,” Charlotte said. ”Letters substituting for an unknown variable is logical. However, there are so many unknowns here. Are we filling these with numbers that can be known when looking at something, or provided?”
“With Cardano’s Formula, the provided numbers are enough to determine the value of X,” Lydia replied. She pointed to Charlotte’s notes, “There are three equations we can use, one for each possible value of X. You can see, each requires use of A and B, but also new variables called S and T. That miniscule i there, do you remember what it represents?”
Charlotte’s hands under the table open and closed as she chewed her lip listening to Lydia and looking at the notes. ”The i is for imaginary numbers.” She had struggled with those, it seemed unreasonable to add something imaginary to maths.
Lydia delivered a curt nod. “S and T represent equations of their own. Each of these equations can be represented in terms of Q and R. Q and R are themselves shorthand for equations which express in terms of A, B, C, and D. This sounds absurd, doesn’t it? Here’s the reason this is done. It’s just breaking it into smaller pieces. One could instead take those first three equations for finding X and expand them to be only in terms of A, B, C, and D. But that would create ungainly monstrosities of numbers which are even harder to keep track of. Instead, let’s follow this example. So Q is equal to 3 times A times C, minus B-squared, and all of that is divided by 9 times A-squared. That is, Q is 3 times 2 times 4, minus 3-squared, all divided by 9 times 2-squared. Simplifying, Q is…24 minus 9—15, divided by 9 times 4—36. Q is 15 divided by 36. Do you follow?”
Charlotte’s feeling of being overwhelmed increased with each moment as Lydia explained. She could recite what Lydia had said, but if asked to explain it differently she could not. ”Can we do this with numbers so I see the process?” Her voice was a bit tight.
“That’s precisely what we’re doing. The formulae are such that one replaces letters with numbers as soon as they become known. So, where we started only knowing the values of A, B, C, and D, now we know Q. The next formula will give us a number with which to replace R. Then, we will have formulae to find S and T. When we have numbers for S and T, we can then find X. One could, instead of going step by step using these smaller equations, combine them all into one large equation. But that would be more confusing. It’s a game of crossing out letters and replacing them with numbers. Does that clarify?”
Charlotte nodded. ”It does.” She still was not excited about the formula, but at least now she might be able to start solving it. Charlotte at least had some experience with playing the crossing out game. This just was a lot longer than normal.
Lydia pointed to the formula for R. “Then, would you be so kind as to replace the letters with their appropriate numbers for this formula?”
Charlotte nodded again and took the pen and paper and started to work. Her stomach twisted a bit as she started working on it and she felt the need to shake her arms. She did twice when she got stuck on particularly annoying bits, but she did not ask Lydia for help. Her goal was to get as far as she could by herself. She would have to get some fresh air after this. On the plus side, everything that had happened between the garden earlier and these numbers and letters had fully driven her brother’s unopened letter from her mind.
Charlotte paused, lifting the pen slightly. She frowned, and went back over her equations. Something was off, but she was not certain what. ”I did something wrong.” She passed the paper to Lydia. ”I cannot find the error.”
Lydia reviewed Charlotte’s work, then produced a second sheet of paper. She replicated Charlotte’s work, did the arithmetic herself, then squinted again at Charlotte’s work. “Arithmetic. That’s all. 2 cubed is 8, and 8 times 54 is…432. Then we reduce. We’ll divide the top and bottom by 2, and call negative 179 divided by 236 good enough. Now, you have Q and R.” Lydia copied down her result onto the main page, then pointed at the equations for S and T. “We can find these values now. Don’t worry so much about the arithmetic. We can do that in our own time. But do you see? S will be the cube root of R, plus the cube root of the sum of Q-cubed and R-squared. T will be the cube root of R, minus the cube root of the sum of Q-cubed and R-squared. It’s a formidable-looking equation, but only three values to fill in for each. That’s all you need to do. Leave the fractions as they are, and replace the letters with their respective values. Now show me you understand what I’m asking of you.”
Charlotte took a slow breath, nodded, and took over the pen and paper again. She hated this. Her body seemed to be slightly offset from where she was supposed to be. It took her a bit longer this time, but she got the equation down to its non-simplified version. She felt more confident at the end of it, though she knew she would have to do several of these as practice prior to her next maths class. ”How is that?” she slid her paper back to Lydia.
“Excellent. Now, we can find X.”
