Solving Quadratic Equations: Finding 'm' Value

Hey guys! Let's dive into a classic math problem. We're going to tackle the equation x² - 6x + m = 0. The main goal is to figure out the different values of m (a real number) that make this equation behave in specific ways. This is super helpful because it really makes you understand how quadratic equations work. We will have to use a bunch of different math methods, and by the end, you'll be able to solve similar problems like a pro. This exploration is going to be super interesting, so let's get started!

Part A: Finding m When x = 2 is a Root

Alright, first things first, let's look at the scenario where x = 2 is a root of the equation. This means that when we plug in x = 2 into the equation, everything should equal zero. It's like finding a secret code – when you use the right key (in this case, x = 2), the lock opens (the equation becomes true). Now, to find m, we're going to substitute 2 for x in our equation and solve for m. It's a simple substitution, but it’s a really important concept in algebra. So, our equation x² - 6x + m = 0 becomes (2)² - 6(2) + m = 0. Let's break this down step-by-step: 4 - 12 + m = 0. Simplify that and you've got -8 + m = 0. Solving for m, we add 8 to both sides, which gives us m = 8. Easy peasy, right? Thus, when m = 8, x = 2 is indeed a root of the equation. Also, We've successfully determined the value of m in this case. The key takeaway here is understanding what it means for a value to be a root of an equation – it's all about making the equation true when that value is plugged in. Now you can solve equations with this information. This is one of the most basic principles in solving algebraic problems, so this is critical.

The Importance of Root Substitution

Why does this method work? Well, a root of an equation is a value that, when substituted for the variable, makes the equation true. In the case of a quadratic equation, the roots are the values of x where the parabola (the shape of the graph of the equation) crosses the x-axis. Knowing this helps us find specific solutions or understand the behavior of the equation. By substituting the root value into the equation, we're essentially using a known point on the graph to help us uncover unknown parts of the equation, like m in our case. It's like having a puzzle piece and using it to find the other pieces that fit in around it. Understanding roots is crucial because it gives us a way to analyze and solve different types of equations. You see this everywhere in higher mathematics.

Part B: Finding m When the Equation Has One Real Solution

Okay, time for the next challenge: when the equation has only one real solution. When a quadratic equation has a single real solution, it means that the parabola touches the x-axis at only one point (the vertex of the parabola). This special case happens when the discriminant (the part under the square root in the quadratic formula) is equal to zero. Remember the quadratic formula? It's: x = (-b ± √(b² - 4ac)) / 2a. The discriminant is b² - 4ac. So, for one real solution, b² - 4ac = 0. In our equation, x² - 6x + m = 0, we have a = 1, b = -6, and c = m. Now, let's plug these into the discriminant: (-6)² - 4(1)(m) = 0. This simplifies to 36 - 4m = 0. Solving for m, we get 4m = 36, and therefore, m = 9. So, when m = 9, the equation has exactly one real solution. Nice work! When m equals 9, the equation turns into x² - 6x + 9 = 0. This is the same as (x-3)² = 0, which means x=3 is the only answer. That is pretty cool, isn't it?

Diving into the Discriminant

The discriminant is a game-changer when it comes to understanding quadratic equations. It tells us how many real solutions an equation has. If the discriminant is positive, the equation has two distinct real solutions. If it’s zero, the equation has exactly one real solution (a repeated root). And if it’s negative, the equation has no real solutions (the solutions are complex numbers). So, the discriminant acts like a traffic light for solutions! By understanding the discriminant, we can predict the behavior of a quadratic equation without actually solving it. This makes it a powerful tool for quickly analyzing equations and choosing the right method to solve them. Knowing the discriminant also helps you understand the shape and position of the parabola on a graph. The discriminant is like a secret code within the quadratic equation and can make complex problems a lot easier.

Part C: Finding m When the Equation Has Two Real Roots

Alright, let's figure out when our equation has two real roots. For this to happen, the discriminant (b² - 4ac) must be greater than zero. This means that b² - 4ac > 0. In our equation, as before, a = 1, b = -6, and c = m. Thus, we get (-6)² - 4(1)(m) > 0, which simplifies to 36 - 4m > 0. Let's solve this inequality: 36 > 4m. Dividing both sides by 4, we get 9 > m, or m < 9. Thus, when m is less than 9, the equation has two real roots. Any value of m less than 9 will result in two distinct points where the parabola crosses the x-axis. Therefore, we found what we wanted.

The Relationship Between Discriminant and Roots

The discriminant helps determine the number and nature of the roots. If the discriminant is positive, the square root in the quadratic formula yields two distinct real values for x, giving us two real roots. These roots represent the x-intercepts of the parabola. The further the discriminant is above zero, the more spread out the roots are on the x-axis. In contrast, when the discriminant is a large negative number, the parabola doesn't intersect the x-axis at all, meaning no real solutions. The roots are complex numbers. By using the discriminant, we can easily determine whether we need to find the roots, without solving the equation. This makes solving problems much more efficient. By understanding this relationship, we can also predict the shape and position of the parabola, and it also makes it easier to work with quadratic equations.

Part D: Finding m When the Equation Has No Real Roots

Last one, guys! Now, we need to find the values of m for which the equation has no real roots. For this to happen, the discriminant must be less than zero (negative). Thus, b² - 4ac < 0. Let's plug in our values again: (-6)² - 4(1)(m) < 0. This simplifies to 36 - 4m < 0. Solving for m: 36 < 4m, and therefore, m > 9. So, when m is greater than 9, the equation has no real roots. This means the parabola doesn't touch or cross the x-axis. All solutions are complex numbers.

Imaginary and Complex Numbers

When the discriminant is negative, we enter the realm of complex numbers. The solutions to the equation are then in the form of a + bi, where a and b are real numbers, and i is the imaginary unit (√-1). In this case, the parabola doesn’t intersect the x-axis. Imaginary numbers are essential in many areas of mathematics and physics. Although we can’t see the roots on the standard real number line, they are mathematically valid solutions, representing the turning points of the graph in an abstract way. Understanding complex numbers enhances your mathematical skills. It gives you the ability to solve equations and problems that extend beyond what's possible with real numbers. Complex numbers are used in electrical engineering, quantum mechanics, and signal processing.

Conclusion: Summarizing the Results

Alright, let's recap what we've learned, since you guys have done such a great job following along! We've successfully navigated the equation x² - 6x + m = 0 under various conditions:

• Part A: m = 8 when x = 2 is a root.
• Part B: m = 9 when the equation has one real solution.
• Part C: m < 9 when the equation has two real roots.
• Part D: m > 9 when the equation has no real roots.

We did this by using the idea of root substitution and really focusing on the discriminant. Using this method, we saw that we can determine the roots of an equation. Using the discriminant, we can classify and analyze them. We've explored the relationship between the discriminant and the nature of the roots. Plus, we've touched on complex numbers. You guys should be proud of your work! I hope you learned a lot and enjoyed the process. Keep practicing, and you'll become quadratic equation wizards in no time!