Finding Roots & Discriminant: A Deep Dive Into Quadratic Functions

Hey math enthusiasts! Today, we're diving deep into the world of quadratic functions, specifically tackling a problem that blends algebra and critical thinking. We'll be exploring the characteristics of a quadratic function, uncovering its discriminant, and figuring out when it has two unique roots. Get ready to flex those brain muscles, because we're about to embark on a mathematical journey that's both challenging and rewarding. Let's get started!

Understanding the Quadratic Function & Its Components

Alright, guys, let's break down the problem step by step. We're given the quadratic function f(x) = 2px² + (2p - 5)x + p - 5/2, where x is a real number (x ∈ ℝ) and p is a rational number (p ∈ ℚ). This function is a polynomial of degree 2, meaning it has the general form ax² + bx + c, where a, b, and c are constants. In our case, a = 2p, b = (2p - 5), and c = p - 5/2. Remember, the values of a, b, and c determine the shape and position of the parabola on the coordinate plane. The coefficient a dictates whether the parabola opens upwards (if a > 0) or downwards (if a < 0), while b and c influence the vertex's location and the y-intercept, respectively. This gives us a solid foundation to work from.

Now, let's talk about the discriminant. The discriminant, denoted by the Greek letter delta (Δ), is a crucial part of a quadratic equation. It's calculated using the formula Δ = b² - 4ac. This seemingly simple formula holds the key to understanding the nature of the roots of a quadratic equation. The discriminant tells us how many real roots the quadratic equation has. If Δ > 0, the equation has two distinct real roots. If Δ = 0, the equation has one real root (or two identical real roots). If Δ < 0, the equation has no real roots (the roots are complex). It's like a crystal ball that reveals the secrets of the quadratic equation's roots! We'll use this formula to solve the first part of the problem. Remember, the values of a, b, and c are important in this function to calculate the discriminant of f(x). Make sure you use the appropriate value in this case.

So, the main goal in this section is to determine the discriminant of the quadratic function f(x). We can get the discriminant by using the formula Δ = b² - 4ac. We already know a, b, and c. By substituting those values into the formula, we can get the discriminant for f(x). This is like putting the pieces of a puzzle together. As we mentioned above, this is an important part of solving the quadratic equation. The discriminant will tell us whether the equation has no real roots, one real root, or two distinct real roots. The value of the discriminant will change based on the value of p.

Calculating the Discriminant of f(x)

Alright, let's get our hands dirty and calculate that discriminant! We know that a = 2p, b = (2p - 5), and c = p - 5/2. Plugging these values into the discriminant formula, we get:

Δ = b² - 4ac

Δ = (2p - 5)² - 4(2p)(p - 5/2)

Now, let's expand and simplify this expression. First, let's deal with (2p - 5)²:

(2p - 5)² = (2p - 5)(2p - 5) = 4p² - 20p + 25

Next, let's expand 4(2p)(p - 5/2):

4(2p)(p - 5/2) = 8p(p - 5/2) = 8p² - 20p

Now, substitute these expanded forms back into the discriminant equation:

Δ = (4p² - 20p + 25) - (8p² - 20p)

Δ = 4p² - 20p + 25 - 8p² + 20p

Finally, let's combine the like terms:

Δ = (4p² - 8p²) + (-20p + 20p) + 25

Δ = -4p² + 25

Boom! We've successfully shown that the discriminant of f(x) is -4p² + 25. High five! We've navigated through the algebraic jungle and arrived at the correct answer. The discriminant, now expressed in terms of p, will tell us about the nature of the roots of our quadratic function.

This is a crucial step in understanding the behavior of our quadratic function. The discriminant, in its final form of -4p² + 25, is now dependent on the value of 'p'. This tells us that the number and nature of the roots of f(x) will change with the variation in the value of p. The beauty of mathematics lies in its ability to reveal such relationships, doesn't it? As we proceed to the next part of the problem, this understanding will be essential.

Finding Values of p for Two Distinct Roots

Now for the fun part: finding the values of p that make f(x) have two distinct roots. Remember, for a quadratic equation to have two distinct real roots, its discriminant (Δ) must be greater than zero (Δ > 0). We've already determined that the discriminant of f(x) is -4p² + 25. Thus, we need to solve the inequality:

-4p² + 25 > 0

Let's rearrange this inequality to make it easier to work with:

4p² < 25

Now, divide both sides by 4:

p² < 25/4

To solve for p, we take the square root of both sides. Remember, when taking the square root of both sides of an inequality, we need to consider both positive and negative roots:

√p² < √(25/4)

|p| < 5/2

This means that p must be within the range of -5/2 and 5/2. We can express this as:

-5/2 < p < 5/2

Since p is a rational number, this inequality tells us the values of p that would make the quadratic equation f(x) have two unique real roots. Any rational number within this range will satisfy the condition. For example, if we have p = 0, then the discriminant would be -4(0)² + 25 = 25, which is greater than 0, meaning it has two unique roots. Another example is p = 2, then the discriminant would be -4(2)² + 25 = 9, which is greater than 0. This further confirms that this calculation is correct.

This is the core concept of this section, guys. We have found the range of p values for which the quadratic equation f(x) has two distinct roots. Understanding how the discriminant impacts the nature of roots allows us to solve various real-world problems. Whether it's in physics, engineering, or even economics, the ability to analyze and solve quadratic equations is a valuable skill. Keep practicing, and you'll find that these mathematical concepts become second nature!

Conclusion: Recap and Key Takeaways

Alright, let's wrap things up. We started with a quadratic function, f(x) = 2px² + (2p - 5)x + p - 5/2, and we showed that its discriminant is -4p² + 25. Then, we figured out that for f(x) to have two distinct roots, p must be within the range of -5/2 < p < 5/2. It's all about connecting the dots, from understanding the components of a quadratic equation to applying the discriminant to determine the nature of the roots. I'm sure you guys will agree that mathematics is really amazing.

Key Takeaways: The discriminant (Δ = b² - 4ac) is the key to determining the nature of the roots of a quadratic equation. If Δ > 0, there are two distinct real roots. If Δ = 0, there's one real root (or two identical real roots). If Δ < 0, there are no real roots. The values of a, b, and c influence the discriminant, and therefore the roots of the equation. Understanding inequalities is crucial when finding the range of values for p. So, keep practicing those inequalities! Also, it's worth noting that the discriminant not only gives information on the types of roots, but can also be used to understand the graphical representation of a quadratic function.

So there you have it, folks! I hope you enjoyed this journey into the world of quadratic functions. Remember, practice makes perfect. Keep exploring, keep questioning, and never stop learning. Until next time, keep those mathematical gears turning!