Heh, thanks so much for the reviews~!
And thank you, Yonaka Takai, for correcting my mistake… xD
Also, there have been votes, so the scoreboard is updated again!
Mukuro/Kaia: 8 [and 5 re-votes from Shin] [one of the votes are from Shoko-Chan, and she re-votes twice]
Mukuro/Kaia with a Hibari twist: 1 [and 3 re-votes from Shoko-Chan]
Gokudera/Kaia: 4 [and 1 1/2 re-votes from Shin] [another re-vote from Crazee4anime]
Hibari/Haruka: 2 [and another re-vote from Shin]
Yamamoto/Chrome: 2 [and 2 re-votes from Shin] [and a re-vote from Banyou]
Hibari/Kaia: 3
Belphegor/Kaia: 2
Soto/Kaia: 1
Byakuran/Kaia: 1
Xanxus/Kaia: 1
Yes, the new added votes include a Hibari/Kaia vote, which is catching up to Gokudera/Kaia! And there is an addition: Xanxus/Kaia. And I do agree with Yonaka Takai; it will be interesting. :D
And that~one, apparently you aren't really alone…there are two other people who voted for that… xD
Thanks, everyone, for reading, too~! :]
Happy Reading, and please review! And vote!
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Dinner was fulfilling to the stomach – everyone was well aware of that. Tsuna retreated back into his room to finish his make-up homework – Gokudera and Yamamoto followed him up into his room.
"Okay! Now – how to factor 2x^2+5x+3!" beamed Gokudera. "Okay, then, let's start with replacing the variable with a random number. Let's start with an easy number, like…one! Okay, then, it would then be 2(1)(1)+5(1)+3, which is 2+5+3, which equals 10. What are the factors of ten, then? That would be… 2 and 5, right? Then- wait… If you factor 2x^2+5x+3 into (2x+3)(x+1), then- yeah, if x=1, and 2x^2+5x+3=10, then 2x+3=5 (which is 2(1)+3=5) and x+1=2 (which is (1)+1=2). Ah, okay. Then, according the zero property of multiplication, either 2x+3 or x+1 must be 0. So then either 2x+3=0 or x+1=0. Then, you would solve for x. Simple, right, Juudaime?"
Much to his surprise, Tsuna was completely lost. Gokudera blinked.
"Sh-Should I write it down on…paper…? It'd probably be easier that way…"
"N-No…that's fine… I think it's better if you do an example for me…so I'd probably figure it out that way…" Tsuna shook his head. "Why wouldn't Reborn help?! He's my tutor, for goodness's sake!!" Gokudera snatched a pad from nowhere and a pencil from nowhere and began to write furiously on the paper. Tsuna feared the paper would rip.
"Well, personally, I'd go by looking at the formula," smiled Yamamoto. "The formula's ax^2+bx+c, right? And if ax^2+bx+c equal these random two factors, they'd have some way of calculating it, right? Well then it would be (a1x+c1)(a2x+c2). If it's that way, then a1×a2=a, and c1×c2=c. Also, a1c2+a2c1=b."
"Gahhhhh!!" Tsuna ran his hands through his hair in exasperation. "I don't get it!!" He looked up. "Even you get it!" Suddenly, Gokudera shoved the pad he had been scribbling on into his Boss's face. Tsuna took it and looked at it. He blinked a few times.
2x^2+5x+3=(2x+3)(x+1)=0
2x+3=0; 2x=-3; x=-3/2
x+1=0; x=-1
2(-3/2)(-3/2)+5(-3/2)+3=9/2-15/2+6/2=9/2+6/2-15/2=15/2-15/2=0
2(-1)(-1)+5(-1)+3=2-5+3=2+3-5=5-5=0
x=-3/2 or -1
"Do you get it now, Juudaime?" smiled Gokudera. Tsuna was overwhelmed. He hadn't a single idea how to factor it – Gokudera didn't help him at all in that part.
"Uhh…" Tsuna daren't say anything else, and Gokudera got the message. He sighed and thought a little more.
"Then…what about this – you multiply these two values-" Gokudera pointed to 2x and x. "-together to get this." He pointed to 2x^2. "You make this and this, multiplied together-" He pointed, this time, to 3 and 1 in the binomials. "-equal this." Gokudera pointed to the 3 in the quadratic. "And you make this, multiplied together-" He pointed to the 3 and the x in the binomials. "-and this multiplied together-" He pointed to the 2x and the 1 in the binomials. "-added together become this." He pointed to 5x. "Get it?"
"Sort of…" frowned Tsuna. He blinked. "Okay, so for number two, which is f(x)=x^2+6x+9, it's about the same thing…?"
"Yeah," nodded Gokudera. "C-Can you do that, Juudaime?"
"Uhh…s-sure, I think…" Tsuna narrowed his eyes and scrunched his eyebrows together in concentration. "So…I have to get two factors of x^2 first, right? Then, it's… either x^2 and 1 or x and x… Okay then, it's going to be x^2 and 1…" He wrote that on his paper.
"Eh- Juudaime, t-that isn't quite right… because i-if you make it x^2 and 1, then how do you get this one out?" Gokudera pointed to the 6x.
"Ohh…" blinked Tsuna. "So that's going to be x and x, right? Okay. Umm… and then… what are the factors of…6x?" Tsuna erased the x^2 and the 1.
"Of 9, Juudaime. Of 9…" blinked Gokudera. Despite himself, he began to think it was hopeless. However, he shoved that thought out of his head – Juudaime saved his life! How could he be hopeless?!
"So, the factors of 9 are… 9 and 1, and 3 and 3, right?" chimed in Yamamoto. Gokudera glared at him.
"R-Right," Tsuna looked down to his paper again. "Then…is it… 9 and 1? Or is it 3 and 3? Wait… 9 and 1 equal 10, and 3 and 3 equal 6, right? So…it's 3 and 3, r-right? Is it?"
"Y-Yeah!!" Gokudera's face lit up. Juudaime got it!!
"Okay…so it's… um… Is it x3 and x3?" Tsuna frowned. "For some reason it looks weird…" Gokudera mentally face-palmed himself.
"It's x plus 3, which is one, solitary value, multiplied by x+3, which is also a single value. Put parentheses around the (x+3)'s."
All three teenagers looked to the door.
"Hi…" Kaia shyly smiled. "I was getting a little bored downstairs, and Reborn said you guys were doing homework. I figured I would go home, but I probably would get lost again." She frowned, and Gokudera's expression fell. Kaia walked over to them, huddled around the low table, and she sat down with them, to Gokudera's right and to Yamamoto's left, directly opposite of Tsuna.
"What are you guys working on, right now?" she smiled. "I think I already finished the quadratics page this afternoon."
"Ah, really?" Tsuna's face lit up with hope. "Then can you help me too?"
"Sure," beamed Kaia. "What do you need help on?"
"Umm…Well…" Tsuna chuckled nervously. "I-I don't know how to factor…"
"Can you multiply binomials though?" blinked Kaia. "If you do, then it's easier – do the opposite, reverse the process."
"Err…I-I don't know how to either…" Tsuna laughed in embarrassment. "C-Can you help me learn it?"
"Sure. You two, too, if you don't know," Kaia addressed both Gokudera and Yamamoto. "I've got a feeling you know, though, Gokudera-san, since you've been explaining that much… I even heard your voice downstairs…"
Gokudera nodded.
"Thought so," huffed Kaia. "Well, I'll give you a simple example. (x+1)(x+1), then." She wrote that down on another page of the paper pad Gokudera had. "There. Now, how do you get that, you ask? There's actually multiple ways. I'll show you one. First, use the distributive property." She wrote on the paper again. "Now, (x+1)(x+1) is x(x+1)+1(x+1). But the one cancels out there." She re-wrote the expression. "x(x+1)+(x+1). Now, use the distributive property again, so it's x^2+x+x+1. And then combine like terms, so x+x is 2x. So the final quadratic is x^2+2x+1. Get it?"
"Ohhhh…" blinked Tsuna. He gave Kaia a grateful smile. "I get it, sort of…" Gokudera gaped. He hadn't a clue how Kaia got that, but it worked, like a miracle. Like how Lambo actually behaved, like… everything.
"Now, about factoring. It's just a big term that refers to the opposite of multiplying binomials like that. It's like division – 3(4)=12=4(3). You multiply it out into the product, and then you convert it back into separate factors. Now, to get that…" Kaia smiled. "I think it's best if you do the multiplying binomials page first." She looked to Gokudera.
"Actually, I didn't hear absolutely everything you've explained, but I think it makes sense too. It's just that…you have to put it in a way that is easy to understand and so that the other person understands perfectly based on previous knowledge…" Kaia beamed. "And in math, you can't skip the rungs in the ladder. Skipping parts to it just brings trouble later, when the rungs get farther and farther away from each other…" Her expression became one of deep thought and longing, as if she were wondrously sophisticated and complex. She shook her head.
"Sorry; did you understand that at all? No, you probably didn't…" Kaia smiled an apologetic smile to the three confused boys. "Sorry."
"It's fine," Gokudera, surprisingly spoke first. "Juudaime, why don't you…get started on that homework? The sooner you finish, the better!!"
"Auuh," nodded Tsuna, and he dug out the paper from the rather formidable pile of make-up homework. He glanced on the first thing he saw. "Ehhh?! The paper explains everything already!!"
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A sad conclusion to Gokudera's efforts to teach Juudaime to do simple algebra. Actually, come to think of it, I wrote this when I, myself, was learning it. Which was in April…? Probably the end of April, when we all get to the f(x) and function stuff. :D
Poor Tsuna this time. Heh. xD
Tune in next time, everyone, and pleae review and vote!
Have an awesome summer [break] everyone! xD
