Finding P - Q + R With Common Root Quadratic Equations
Hey guys! Let's dive into a cool math problem today that involves quadratic equations and finding a specific value. We're going to break down the problem step by step, so you can totally ace it. Our main goal here is to figure out the value of the expression (p - q + r) given two quadratic equations that share a common negative root. Sounds intriguing, right? So, let’s get started and make sure to really nail those keywords early on to help everyone understand what we’re up to!
Understanding the Problem
Okay, so we have two quadratic equations:
- px² + qx + r = 0
- rx² + qx + p = 0
It's super important to note that p isn't equal to r, and neither of them is zero (p ≠r ≠0). This little detail is a biggie because it tells us we're dealing with two different quadratic equations. The key here is that these two equations have a common negative root. This means there's a negative number that, when plugged into both equations for x, makes both equations true. Our mission, should we choose to accept it (and we do!), is to find the value of the expression (p - q + r). We need to use all the information that has been provided to ensure we get to the correct answer. Think of it like solving a puzzle – every piece of information fits together to reveal the solution. So, what's our game plan? First, we'll use the fact that the equations share a common root to create a relationship between p, q, and r. Then, we'll use that relationship to calculate the value of (p - q + r). Remember, in math, sometimes the trickiest problems can be solved by just breaking them down into smaller, more manageable steps. Stay with me, and we'll conquer this together! Understanding the problem is the first step, and now we're ready to roll up our sleeves and dive into the solution.
Setting Up the Equations with the Common Root
Alright, let's get our hands dirty with some algebra! Since both quadratic equations share a common negative root, let's call that root -α, where α is a positive number (because we know the root is negative). This is a crucial step, guys, because it allows us to actually plug something concrete into our equations. Now, because -α is a root of both equations, it means that if we substitute x with -α in both equations, they should both equal zero. This is the fundamental idea behind solving this kind of problem, and it's something you'll use again and again in math.
So, let’s do it! Plugging -α into our first equation, px² + qx + r = 0, we get:
p(-α)² + q(-α) + r = 0
Which simplifies to:
pα² - qα + r = 0 (Equation 1)
Now, let's do the same for the second equation, rx² + qx + p = 0. Substituting x with -α, we have:
r(-α)² + q(-α) + p = 0
Which simplifies to:
rα² - qα + p = 0 (Equation 2)
We've now got two equations with three unknowns (p, q, r, and α), but don't panic! This is perfectly normal in these types of problems. The trick is to use both equations together to eliminate one of the unknowns or find a relationship between them. This is where our problem-solving skills really come into play. We’ve set up the equations, and now we need to find a way to combine them to get to our goal: finding the value of (p - q + r). So, what’s our next move? Stick with me, and we'll figure it out!
Solving for the Relationship Between p, q, and r
Okay, so we've got our two equations:
- pα² - qα + r = 0
- rα² - qα + p = 0
Our next step is to find a relationship between p, q, and r. Notice that both equations have a -qα term. This is a fantastic opportunity! Why? Because it means we can eliminate that term by subtracting one equation from the other. Elimination is a super handy technique in algebra, and it's exactly what we need here to simplify things.
Let’s subtract Equation 2 from Equation 1. This gives us:
(pα² - qα + r) - (rα² - qα + p) = 0 - 0
Now, let’s simplify this beast. Open up those parentheses and combine like terms:
pα² - qα + r - rα² + qα - p = 0
Notice that the -qα and +qα terms cancel each other out! Hallelujah! That simplifies things beautifully. We're left with:
pα² + r - rα² - p = 0
Now, let's rearrange the terms to group similar variables together:
pα² - rα² - p + r = 0
We can factor out an α² from the first two terms and rearrange the last two terms:
α²(p - r) - (p - r) = 0
Now we see a common factor of (p - r)! Let’s factor that out:
(p - r)(α² - 1) = 0
Remember at the beginning of the problem, we were told that p ≠r. This is crucial because it means (p - r) cannot be zero. If (p - r) were zero, then p would equal r, which contradicts the given information. So, what does that mean for the other factor, (α² - 1)? It must be zero!
α² - 1 = 0
This gives us:
α² = 1
Taking the square root of both sides, we get α = ±1. But remember, we said α is a positive number because our common root is negative (-α). So, we can conclude that:
α = 1
We've made a huge breakthrough! We now know the value of α. This is a major piece of the puzzle. Now that we know α, we can use it to find the relationship between p, q, and r. We're on the home stretch now – let’s keep pushing!
Calculating the Value of (p - q + r)
Awesome work, guys! We've figured out that α = 1. Remember, -α is our common negative root, so our common root is x = -1. Now, we can use this information by plugging x = -1 into either Equation 1 or Equation 2 (from our earlier steps) to find a relationship between p, q, and r. Let’s use Equation 1:
pα² - qα + r = 0
Since α = 1, we have:
p(1)² - q(1) + r = 0
Which simplifies to:
p - q + r = 0
BOOM! We've done it! We've found the value of (p - q + r). It’s zero!
So, if the equations px² + qx + r = 0 and rx² + qx + p = 0 (p ≠r ≠0) have a negative common root, then the value of (p - q + r) is equal to 0. This corresponds to option (3) in the original problem. This whole problem-solving process can be applied to multiple math equations in the future.
Conclusion
Isn't math cool? We started with a seemingly complex problem involving quadratic equations and a common root, and we broke it down step by step until we found the solution. We used the information provided, set up equations, eliminated variables, and finally calculated the value of (p - q + r). The key takeaways here are:
- Understanding the problem and identifying the key information.
- Setting up equations based on the given information.
- Using algebraic techniques like substitution and elimination to solve for unknowns.
- Paying close attention to details and constraints (like p ≠r ≠0).
Most importantly, remember that problem-solving is a skill that gets better with practice. So, keep challenging yourselves, keep breaking down problems into smaller steps, and you'll be amazed at what you can achieve! You guys nailed it! This exercise was a great way to improve our math skills, right? Keep practicing, and you'll become math superstars in no time! Remember to always focus on the keywords and information given. Great job, everyone!
