"What?" Ron Weasley exclaimed.
"You heard me Weasley." Professor Babbage replied.
The sixth year Hogwarts students had just gotten their latest Muggle Studies Assignment; to learn all about 10th grade math in the Muggle World. This class's unit? Quadratic Equations. Many students turned their heads in confusion at the word.
"What's a quad – dra – tick?" One student said.
"Is it a disease?" Another yelled out.
"No no no no NO! It is a mathematical function muggles use in daily life to help determine things such as how far a ball was thrown, or the trajectory height of a rocket. Things we would normally use magic to solve." Babbage explained.
Harry Potter sighed. He hadn't heard of such equations ever since he had left the muggle world for the wizarding world. The most he ever heard of was the equation y=x2 + bx + c, and even that boggled his mind. Sheets were being passed out, each one with a horribly complex equation on it.
"Oh Merlin's Beard…." Ron moaned. "What the hell does this mean!"
Harry stared at the question:
There is a Quidditch game. The keeper throws the quaffle in the air. Its height, in meters and its time, in seconds can be measured with the equation H = - 4.9t2 + 38t + 1.75. What time will the chaser, who is 50 meters in the air, catch the quaffle? What is the maximum height?
Harry blinked, hoping it was all a dream. The sheet of paper was still there. The question was still there. Everything was real. The question was real. The horrible, complex, too many numbered question. Harry picked up his quill and dipped the end gingerly in ink. He poised his hand above the page, and paused. A droplet of ink splashed on the page. Who was he kidding? He had never learnt this, he was a wizard for God's sake!
"Psst!" A tiny voice said. Harry looked around.
"Wow, you really are quite thick, down here!" The voice said again. Harry looked down at his quill. Two eyes were bulging from it, and a mouth was moving.
"Hey! Do you need help with your work?" Since Harry was a wizard, he was only mildly surprised at his quill.
"Ya, I have no idea what this is!" Harry exclaimed.
"Ha! That's simple! This equation is a quadratic equation in standard form!" His quill talked with ease.
"A what in what form?" Harry tried to keep up.
"A quadratic formula in standard form. This equation can help you determine all the answers to the questions above, provided you do the correct steps. I'll walk you through it!" His quill said happily.
"Alright…" Harry didn't know whether to trust his quill, but what choice did he have? "How do I find out the first question?"
"The first one is easy. The chaser is 50 meters high. Since the equation is H = - 4.9t2 + 38t + 1.75, that means that H, the height, is 50 meters right?"
"Right…"
"So just plug in the 50 to where the H is, making it 50 = - 4.9t2 + 38t + 1.75. Then you would move the 50 to the other side. I assume you know what happens to the 50?" his quill inquired.
"Ya, the 50 becomes negative. Does that mean the last term would become 1.75 – 50 which would be -48.25?" Harry slowly asked.
"Yup! So now, your equation is 0 = - 4.9t2 + 38t – 48.25. Now, we're going to do something called 'common factoring'. Ever heard of it?" The quill asked.
"Er, no." Harry replied, feeling quite stupid.
"Should'a known. Well, basically, we try to find the Greatest Common Factor or GCF of all three terms. In this case, it would only be – 1, making our equation 0 = - (4.9t2 - 38t + 48.25). Now can you try to guess what goes on next?"
"Well, if I remember correctly, there's something called the quadratic formula involved?" Harry said unsure.
"Yes! This equation cannot be factored as a simple trinomial, because t2 does not equal one, and I'd like to see you try and find two numbers that add up to – 38 and multiply to 236.425! No complex trinomials either. So we use something called the quadratic formula, which goes like:
x = -b +- (square root)b2 -4ac/2a
"Whoa whoa whoa! That equation is…is so….hard!" Harry exclaimed.
"But it works. See, now that we have this formula, we simple plug in our numbers. 4.9 = a, -38 = b and 48.25 = c. So you would get this..." The quill quickly jotted down the two equations Harry would need to solve.
"Okay so now, try to figure it out!" Harry's quill said excitedly.
Harry struggled. He couldn't remember much from when he was living in the muggle world. But he took it step by step. First, he solved what he knew was the discriminant, or b – 4ac. He eventually wound up with 498.3. Then, he figured out that 2(4.9) was 9.8. Having solved that, he continued on with solving two equations:
Harry soon figured out that the two possible times the chaser could have caught the quaffle was either 6.15 seconds, or at 1.6 seconds. Elated, he allowed his quill to correct his work.
"Nice job! But don't get cocky, you still have to find the maximum height of the quaffle!" The quill teased.
"The maximum height? Oh lord…" Harry sighed, his head hurting with all the numbers presented to him. "When will I ever need to know this, might I ask?"
"Well, for one thing, you could use it as a strategy plan while playing Quidditch. Secretly program the quaffle to correspond with a quadratic equation and all you must do is know at what height to catch it or how far away you should be to catch it properly." The quill answered.
"All right, then, let's just get this over with." Harry muttered.
"So, first, you are going to take your equation, H = - (4.9t2 - 38t - 1.75), and we are going to try and convert in to something called vertex form. This form is going to help us find the vertex of the parabola, or in this case, the maximum height of the quaffle." Harry' s quill said.
"Okay, so do we factor again?"
"NO! Jeez, what we want to do is to complete the square."
"Er, complete the square?" Harry was confused.
"Yup, so you put brackets around the first two terms and-"
"So it would be H = - [(4.9t2 - 38t) - 1.75]." Harry said slowly.
"Correct. Now let me finish! So now, you are going to factor out 4.9 to make t2 equal to 1. So now it will look like H = - [4.9(t2 – 7.76t) - 1.75]. Once you get this, you will use the equation (b/2)2 to make it a perfect square. And that's what completing the square is!" Harry's quill said excitedly.
"Ok, so (-7.76/2)2 is equaled to…15.1. Would that make my equation H = - [4.9(t2 – 7.76t + 15.1) - 1.75]?"
"Close, but you must remember to also subtract 15.1 and multiply it by 4.9 when taking it out of the bracket, or you're changing your equation! So it is really H = - [4.9(t2 – 7.76t + 15.1) - 1.75 - 74]."
"Oh! So now the equation (t2 – 7.76t + 15.1) is a perfect square! That means it is equaled to (t – 3.88)2!" Harry exclaimed happily.
"Yup! Now your equation is H = -4.9(t – 3.88)2+76.75! Therefore, the quaffles maximum height is 76.75 meters. Man, this keeper must have a really good throw!" His quill said. "Now, your all done! What have you learned about quadratics today Harry?"
"Well, I guess it seemed complicated at first! But now I know about the quadratic formula, and how it is only used when you can't factor your equation anymore to solve for your x-intercepts. And also, when trying to find the maximum, or minimum too, you must complete the square to find the vertex form! Wow! I never thought I'd remember all of this! Thanks quill!" Harry said.
"No problem. There is one small deal we must complete though…"His quill said slyly.
"What?"
"I MUST TAKE YOUR SOUL." His quill grew to 30 times its original size, and opened its mouth wide.
"AAAAHHHH!" Harry screamed as he awoke with a start. He was sitting upright in his bed, his body covered in a cold sweat. What a terrible dream! Honestly, this was the last year he would be taking Muggle Studies! Harry got dressed in his Quidditch uniform and went down to the Great Hall for some toast. He made his way to the Quidditch pitch and got ready for the big game against Slytherin. He greeted his fellow teammates.
"So what's the plan Harry?" Ginny asked. "Where do you want us to be to catch the ball? How high should we fly today?"
Harry thought for a bit. He paced the tent. He stroked his chin. Finally, smiling, he tapped the projector screen with his wand.
"Okay guys. Time for a crash course called Quadratic Quidditch…."
Sorry for all the equations :S the big two beside the t is supposed to be squared...FF won't let me properly do a squared. Hope you like it!