Finding Roots: When Does Px²-4x+1 Avoid Equal Solutions?

Hey guys! Let's dive into a cool math problem today. We're going to explore the quadratic function f(x) = px² - 4x + 1 and figure out the values of p (where p definitely isn't zero, got it?) that make this function not have equal roots. Basically, we want to know when the function's graph doesn't just barely touch the x-axis at one point. This kind of problem is super important in understanding how quadratic equations work, and it's a fundamental concept in algebra. Let's break it down step by step, making sure we get the full picture.

Understanding Quadratic Equations and Their Roots

First off, let's get on the same page about quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually x) is 2. The general form is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The roots of a quadratic equation are the values of x that satisfy the equation – in other words, the points where the graph of the function crosses the x-axis. These roots are also sometimes called zeros or solutions.

Now, the discriminant is our best friend here. The discriminant (often denoted as Δ or the Greek letter delta) of a quadratic equation is given by the formula: Δ = b² - 4ac. This little formula tells us a lot about the nature of the roots. Here's the lowdown:

• If Δ > 0: The equation has two distinct real roots. This means the graph of the function crosses the x-axis at two different points.
• If Δ = 0: The equation has one real root (or two equal real roots). The graph touches the x-axis at exactly one point – the vertex of the parabola.
• If Δ < 0: The equation has no real roots. The graph does not cross the x-axis; the roots are complex.

So, for our problem, we want to figure out when f(x) = px² - 4x + 1 does not have equal roots. This means we want to find out when the discriminant isn't equal to zero. Understanding the discriminant is absolutely critical for solving problems like this, because it directly links the equation's coefficients to the nature of its roots. Without knowing the discriminant, you'd be stumbling in the dark! Therefore, the crucial part is to correctly apply the discriminant formula and understand how to interpret its results.

Applying the Discriminant to Our Problem

Alright, let's get down to business. We've got f(x) = px² - 4x + 1. Comparing this to the general form ax² + bx + c, we can see that a = p, b = -4, and c = 1. Remember that we're told p ≠ 0, which is important. This ensures we're dealing with a quadratic and not something simpler. We can now use the discriminant formula:

Δ = b² - 4ac

Substitute the values:

Δ = (-4)² - 4 * p * 1

Simplify:

Δ = 16 - 4p

We want to find the values of p for which f(x) does not have equal roots. Therefore, we want Δ ≠ 0. So, let's set up the inequality:

16 - 4p ≠ 0

Now, solve for p:

16 ≠ 4p

p ≠ 16 / 4

p ≠ 4

So, there you have it! The condition for f(x) = px² - 4x + 1 not to have equal roots is p ≠ 4. This means that as long as p is anything except 4, the quadratic equation will either have two distinct real roots or no real roots (complex roots). This analysis is key to answering the original question about avoiding equal roots. We've gone from the general concept of the discriminant down to a specific solution tailored to our equation. Isn't that neat?

Interpreting the Results

Let's really think about what this means. If p = 4, then Δ = 0, and we get a single, repeated root. The graph of the parabola would touch the x-axis at just one point. If p is less than 4, then Δ > 0, and we get two distinct real roots. The parabola crosses the x-axis twice. And if p is greater than 4, then Δ < 0, and we get no real roots. The parabola doesn't touch the x-axis at all. It floats above or below it. This shows us the intimate relationship between the value of p and the behavior of the quadratic function's roots.

Imagine the graph of f(x) as p changes. As p approaches 4 from below, the two roots get closer and closer together until they merge at a single point when p = 4. Then, as p becomes greater than 4, the roots disappear into the realm of complex numbers, and the parabola lifts off the x-axis. Pretty visual, right? This analysis is what makes algebra and especially quadratic equations, so fascinating! This entire process allows us to understand how changes in the coefficients affect the roots and the overall behavior of the function. It's a fundamental concept in mathematics that has applications in physics, engineering, and many other fields.

Conclusion

So, there you have it, folks! We've successfully determined that for the function f(x) = px² - 4x + 1 not to have equal roots, p must not equal 4 (p ≠ 4). This process highlights the importance of the discriminant in determining the nature of the roots of a quadratic equation. We've also seen how a single parameter, p, can dramatically change the behavior of the function. We started with the basic concepts of quadratic equations, the role of the discriminant, and the relationship between coefficients and roots. Then, we applied these concepts to our specific problem. And finally, we interpreted our results to understand how the values of p impact the roots of the equation. This kind of systematic problem-solving approach is crucial for mastering any math concept. Keep practicing, keep questioning, and you'll be acing these problems in no time!

I hope this explanation was helpful. If you have any questions, feel free to ask! Understanding the discriminant, how it affects the roots of a quadratic equation, and how to apply it in solving problems like these is absolutely critical. Good luck, and keep up the great work!