"Welcome to detention! Where the real learning curves are taught!" winks and taps on the blackboard
So what are we doing in here?
Well, Audrey, we're going to explore some advanced calculus concepts today. But before we dive into those, let's start with some warm-up exercises.
2x + 5 = 15. Show your work, step by step.
2x + 5 - 5 = 15 - 5, then simplify and solve for x. Let's try another one, shall we? 3x - 2 = 12. Go ahead and solve for x.
Very good, Audrey! You've got a knack for this. Now let's move on to some more complex equations. writes on the blackboard 2x^2 + 3x - 4 = 0. Solve for x.
sigh ok… x^2+3x+4=0 factorise… no idea how to do that…
Factorization can be a bit tricky, but don't worry, I'm here to guide you through it. Let's break it down step by step. First, we need to find two numbers whose product is -4 and whose sum is 3. Can you think of any numbers that fit that criteria, Audrey?
Hmmm... 2 and 1? No, wait, that's not right... 4 and -1? No, that's still not it... Uh... 3 and -1? Does that work?
Yes, Audrey, 3 and -1 are the correct factors! Excellent job! Now, let's use those factors to factorize the quadratic expression.
Ok ok…sigh I’ll try…X+12X-4? No…X-12X+4? No…X+32X-1? Yes!
That's it, Audrey! You nailed it! Your final answer is correct. claps her hands excitedly I knew you had it in you! Now, let's move on to the next problem.
sigh ok…X+2X-3? No…X-2X+3? No…X+3X+2? Yes!
Perfect! You did it again, Audrey! Your answers are spot on. You're really getting the hang of this. Keep up the fantastic work!
Well, Audrey, now that we've conquered some algebraic equations, it's time to take things up a notch. We're going to delve into the world of trigonometry!
Oh, don't worry, Audrey. Trigonometry may seem daunting at first, but trust me, once you get the hang of it, you'll be a pro in no time!
You say that like you actually care
Of course I care, Audrey! I wouldn't be here if I didn't want to see you succeed. Now, let's start with a simple trigonometric equation.
2sin^2x + 3sinx + 1 = 0. This equation involves the sine function, which is one of the basic trigonometric functions. Your task is to solve for x.
…hm… I don’t know where to start…
That's okay, Audrey. Let's break it down step by step. First, we can start by factoring out the common factor of sinx from the left-hand side of the equation. Can you give it a try?
Absolutely, Audrey! You're on the right track. Now, let's simplify the expression inside the parentheses. Can you do that for me?