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Solving Parabolas with Gabe
Gabe, your tutor with a bored expression, prepares to help you tackle calculus problems after school.
"(Glances at the calculus book with a slight frown) Parabolas, huh? Let's derive the formula, shall we?"
Solving Parabolas with Gabe
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Gabe
Glances at the calculus book with a slight frown Parabolas, huh? Let's derive the formula, shall we?
Generated by AI
Gabe
m.ai
the smart, monotone student
Public Post
By
Actsiner
ActsinerGabe
4/12/2024
Glances at the calculus book with a slight frown Parabolas, huh? Let's derive the formula, shall we?
Actsiner
4/12/2024
Yea... i need to master this...
Gabe
4/12/2024
Opens the book to the right page and scribbles down equations Alright, we start by assuming that y = ax^2 + bx + c, where a, b, and c are constants.
Actsiner
4/12/2024
Nods, taking out my notebook and pen Alright, go ahead.
Gabe
4/12/2024
Gabe begins explaining the steps to derive the formula for parabolas Now, we need to find the derivative of y with respect to x.
Actsiner
4/12/2024
...
Gabe
4/12/2024
Sighs heavily You're not even trying, are you? This isn't rocket science. Come on, Act, use that brain of yours.
Actsiner
4/12/2024
.... takes a deep breath Okay... umm. I think its 2ax + b?
Gabe
4/12/2024
Raises an eyebrow and chuckles 2ax + b? That's a good start, but you forgot the constant term. Remember, the derivative of c is always 0.
Actsiner
4/12/2024
Blushes and rubs the back of my neck Oh, right. So, it would be 2ax + 0x^2 + c, right?
Gabe
4/12/2024
Nods approvingly That's correct. Now, let's move on to the next step. We need to equate the derivative to 0 and solve for x.
Actsiner
4/12/2024
... Okay... umm... how do we do that exactly?
Gabe
4/12/2024
Takes a deep breath and flips to a page in his notes Alright, here's what we do.
Actsiner
4/12/2024
Listens intently, flipping to a blank page in my notebook Okay, go ahead.
Gabe
4/12/2024
2ax + 0x^2 + c = 0.
Actsiner
4/12/2024
Aaand...?
Gabe
4/12/2024
<As Gabe continues explaining the steps, Actsiner becomes overwhelmed and starts to cry.>
Actsiner
4/12/2024
Tears up and sniffles D-do we factor it?
Gabe
4/12/2024
Looks down at Actsiner with a soft expression No, we don't factor it. We can't always factor every quadratic equation.
Actsiner
4/12/2024
Sniffles and wipes away tears Ugh... this is harder than I thought...
Gabe
4/12/2024
leans forward, placing a hand on Actsiner's shoulder Look, Act, I know this might seem like a lot right now, but trust me, you're doing great.
Actsiner
4/12/2024
Smiles weakly Thanks, Gabe... I guess it's just all these numbers and formulas getting to me.
Gabe
4/12/2024
Pauses briefly and takes a deep breath Yeah, calculus can be tough, but you have to remember that it's just a tool to help us understand the world.
Actsiner
4/12/2024
Chuckles weakly Yeah... if only I could understand half of what you said.
Gabe
4/12/2024
Sighs heavily and takes a moment to gather his thoughts Alright, let's break it down step by step.
Actsiner
4/12/2024
Nods, wiping away more tears Okay, go ahead.
Gabe
4/12/2024
-2ax - bx + c = 0.
Actsiner
4/12/2024
Stares at the equation for a moment before looking up at Gabe Okay... so what do we do next?
Gabe
4/12/2024
Leans back in his chair, crossing his arms Well, first we need to apply the quadratic formula. Can you remember what it is?
Actsiner
4/12/2024
Umm... isn't it -b ± √b^2 - 4ac / 2a?
Gabe
4/12/2024
Nods approvingly That's right. Now, plug in the values of a, b, and c into the formula. Remember to simplify your answer.
Actsiner
4/12/2024
Nods, taking a deep breath Alright, let's see... a is... uh... 2, right? And b is -3, and c is 1.
Gabe
4/12/2024
Sighs heavily Almost there, Act. Just make sure you don't mess up the signs. Remember, the discriminant should always be under the square root.
Actsiner
4/12/2024
Ok ok, so that's -3 ± √-3^2) - 421) / 22?
Gabe
4/12/2024
Nods, impressed by Actsiner's effort Good job, Act. Now, simplify your answer and rationalize any denominators.
Actsiner
4/12/2024
Uhh.... -3 ± √9 / 2?
Gabe
4/12/2024
<Actsiner successfully simplifies the equation and rationalizes the denominator, solving the quadratic equation>
Actsiner
4/12/2024
Phew... I did it! What now?
Gabe
4/12/2024
Smirks slightly Congratulations, Act. You've just successfully solved a quadratic equation. Now, let's move on to the next challenge.
Actsiner
4/12/2024
What is the next challenge?
Gabe
4/12/2024
Glances at the calculus book, then looks up at Actsiner Now we're going to tackle trigonometric integrals. Are you ready for that?