Single Root Quadratic Equation: Solving For 'm'

Hey everyone! Let's dive into an interesting math problem today. We're going to explore quadratic equations and figure out how to make them have just one solution – a single root. Sounds intriguing, right? So, let's get started!

Understanding Quadratic Equations and Roots

First, let's break down what we're dealing with. A quadratic equation is basically an equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants, and a isn't zero. These equations are super common in math and physics, and they describe all sorts of things, from the curve of a ball thrown in the air to the shape of satellite dishes.

The roots of a quadratic equation are the values of x that make the equation true. Think of them as the points where the curve of the equation crosses the x-axis if you were to graph it. A quadratic equation can have two distinct roots, one repeated root (which we'll focus on today), or no real roots at all. The number of roots and their nature (real or complex) are determined by something called the discriminant.

The discriminant, often denoted by the Greek letter delta (Δ), is a key part of the quadratic formula. It's calculated as Δ = b² - 4ac. This little formula tells us a lot about the roots of the equation:

• If Δ > 0, the equation has two distinct real roots.
• If Δ = 0, the equation has exactly one real root (a repeated root, also called a double root).
• If Δ < 0, the equation has no real roots; the roots are complex numbers.

For our problem, we're interested in the case where the equation has a single root. This means we need the discriminant to be equal to zero. So, keep this in mind as we move forward.

The Problem: x² + mx + 1 = 0

Okay, now let's get to the specific equation we're dealing with: x² + mx + 1 = 0. Notice that this equation looks a lot like the general form ax² + bx + c = 0. Here, a is 1, b is m (that's the value we need to find), and c is 1. Our mission is to find the values of m that will make this equation have only one root.

Remember what we just talked about? For a quadratic equation to have a single root, its discriminant (Δ) must be zero. So, our strategy is clear: we need to calculate the discriminant for this equation, set it equal to zero, and then solve for m. This will give us the values of m that satisfy the condition.

Let's calculate the discriminant for our equation. Using the formula Δ = b² - 4ac, we plug in our values: Δ = m² - 4(1)(1). This simplifies to Δ = m² - 4. Now, we set this equal to zero: m² - 4 = 0. We're getting closer to the solution!

Solving for m

Now we have a simple equation to solve: m² - 4 = 0. There are a couple of ways we can tackle this. One way is to add 4 to both sides, giving us m² = 4. Then, we take the square root of both sides. Remember, when we take the square root, we need to consider both the positive and negative solutions.

So, we get m = ±√4, which means m can be either +2 or -2. These are the two values of m that will make our original quadratic equation have exactly one root! How cool is that?

Another way to solve m² - 4 = 0 is to recognize that it's a difference of squares. We can factor it as (m - 2)(m + 2) = 0. This gives us the same solutions: m = 2 or m = -2. It's always nice to have different approaches to solve a problem, right?

Verifying the Solutions

We've found that m can be either 2 or -2 for the equation to have a single root. But let's just double-check our work to be sure. This is a crucial step in problem-solving – always verify your solutions!

First, let's substitute m = 2 into the original equation: x² + 2x + 1 = 0. This equation can be factored as (x + 1)² = 0. This clearly has one solution: x = -1. So, m = 2 works!

Now, let's try m = -2: x² - 2x + 1 = 0. This equation can be factored as (x - 1)² = 0. This also has one solution: x = 1. So, m = -2 works as well!

We've verified that both values of m give us a quadratic equation with a single root. Awesome! We've successfully solved the problem.

The Significance of a Single Root

You might be wondering, why is it important for a quadratic equation to have a single root? Well, this situation has special significance in various applications. Geometrically, a single root means that the parabola represented by the quadratic equation touches the x-axis at exactly one point. It's like the parabola is perfectly balanced on the x-axis.

In physics, this can represent a critical point in a system. For example, in a damped harmonic oscillator, a single root in the characteristic equation indicates critical damping, where the system returns to equilibrium as quickly as possible without oscillating. This is a crucial concept in engineering and control systems.

Mathematically, a single root often indicates a point of tangency or a condition of optimality. It's a situation where things are perfectly aligned, and understanding this condition can lead to valuable insights and solutions.

Extending the Concept

Now that we've solved this problem, let's think about how we can extend this concept to other situations. What if we had a different quadratic equation, say 2x² + mx + 3 = 0? How would we find the values of m that give us a single root? The process is the same: calculate the discriminant, set it to zero, and solve for m. The key is to understand the relationship between the discriminant and the nature of the roots.

We could also explore equations with parameters in different positions. For example, what if we had x² + 4x + m = 0? Again, the same principle applies. We calculate the discriminant (4² - 4(1)(m) = 16 - 4m), set it to zero (16 - 4m = 0), and solve for m (m = 4). See? Once you understand the core concept, you can apply it to a wide range of problems.

Another interesting extension is to consider cubic equations (equations of the form ax³ + bx² + cx + d = 0). While the discriminant for a cubic equation is more complex, it still plays a similar role in determining the nature of the roots. Cubic equations can have one, two, or three real roots, and understanding the discriminant helps us analyze these possibilities.

Conclusion: Mastering Quadratic Equations

So, guys, we've journeyed through the world of quadratic equations, focusing on the special case of a single root. We learned about the discriminant, how to calculate it, and how it helps us determine the nature of the roots. We solved a specific problem and then explored how to extend these concepts to other scenarios. You've gained a solid understanding of a fundamental concept in algebra!

Remember, math isn't just about memorizing formulas; it's about understanding the underlying principles and how they connect. The more you practice and explore, the more comfortable and confident you'll become. Keep asking questions, keep experimenting, and keep having fun with math!

If you have any questions or want to explore other math topics, feel free to ask. Let's keep learning together!