Solve 2x² + X + 4 = 0: A Step-by-Step Guide

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Hey everyone! Today, we're diving into the fascinating world of quadratic equations. Specifically, we're going to help Jonathan solve the equation 0 = 2x² + x + 4. This is a classic example of a quadratic equation, and we'll walk through the steps to find its solutions. If you've ever felt lost when facing these equations, don't worry, you're in the right place! We'll break it down in a way that's easy to understand, even if you're just starting your math journey. So, let's put on our thinking caps and get started!

Understanding Quadratic Equations

Before we jump into solving Jonathan's equation, let's quickly recap what a quadratic equation is. In simple terms, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is:

ax² + bx + c = 0

Where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (if 'a' were zero, it would be a linear equation, not a quadratic). These constants determine the shape and position of the parabola that represents the equation when graphed. The solutions to a quadratic equation, also known as roots or zeros, are the points where the parabola intersects the x-axis. These solutions tell us the values of 'x' that make the equation true. Now, there are a few ways to find these solutions, and we're going to use one of the most reliable methods today: the quadratic formula.

Why the Quadratic Formula is Our Go-To Tool

Okay, so why are we making a big deal about the quadratic formula? Well, guys, it's a powerhouse! The quadratic formula is a universal tool for solving quadratic equations, no matter how messy they look. It's especially useful when the equation doesn't factor easily, which is often the case in exams and real-world problems. Factoring, while a neat trick when it works, can be time-consuming and tricky if the numbers aren't playing nice. Completing the square is another method, but it involves a bit more algebraic manipulation, which can lead to mistakes if you're not super careful. The quadratic formula, on the other hand, is a direct route to the solutions. You plug in the values, do the math, and bam! You've got your answers. It's like having a cheat code for quadratic equations, but without actually cheating, of course! Plus, it's super consistent. Whether your equation has nice, whole-number solutions or crazy, complex ones, the quadratic formula will handle it with grace. So, if you're looking for a method that's reliable, efficient, and gets the job done, the quadratic formula is your best friend in the world of quadratic equations. Let's dive into what it looks like and how we use it!

The Mighty Quadratic Formula

The quadratic formula is a mathematical expression that provides the solutions to any quadratic equation in the standard form ax² + bx + c = 0. This formula is derived by using the method of completing the square, but we won't go into that derivation today. Instead, let's focus on what the formula actually is and how to use it. Drumroll, please... Here it is:

x = (-b ± √(b² - 4ac)) / (2a)

Now, I know it might look a bit intimidating at first glance, but trust me, it's not as scary as it seems. Let's break it down piece by piece:

  • x: This represents the solutions (or roots) of the quadratic equation.
  • -b: This is the negative of the coefficient 'b' in the quadratic equation.
  • ±: This symbol means "plus or minus." It indicates that there are two possible solutions: one where you add the square root and one where you subtract it. This is because quadratic equations can have up to two distinct solutions.
  • √(b² - 4ac): This is the square root of the discriminant (we'll talk more about the discriminant in a bit). 'b² - 4ac' is the part under the square root sign, and it plays a crucial role in determining the nature of the solutions.
  • 2a: This is twice the coefficient 'a' in the quadratic equation.

The formula essentially takes the coefficients 'a', 'b', and 'c' from our quadratic equation and plugs them into a specific arrangement that spits out the solutions. The beauty of this formula is its ability to handle any quadratic equation, regardless of the complexity of the coefficients. It works whether the solutions are real numbers, imaginary numbers, or a combination of both. Once you get the hang of identifying 'a', 'b', and 'c', plugging them into the formula becomes second nature. It's like following a recipe – you have your ingredients (the coefficients), you have your instructions (the formula), and you end up with a delicious result (the solutions!).

Unpacking the Discriminant: The Key to Understanding Solutions

Alright guys, let's zoom in on a super important part of the quadratic formula: the discriminant. Remember that part under the square root, b² - 4ac? That's the discriminant, and it's like a secret decoder for the solutions of our quadratic equation. The discriminant tells us a lot about the nature of the solutions without us even having to fully solve the equation. Think of it as a mathematical weather forecaster, predicting what kind of solutions are headed our way.

  • If b² - 4ac > 0 (positive): This means we'll have two distinct real solutions. Real solutions are the everyday numbers we're used to dealing with – like 2, -3, 1.5, etc. In terms of the graph, this means the parabola will intersect the x-axis at two different points.
  • If b² - 4ac = 0: This means we'll have exactly one real solution (which is sometimes called a repeated root). Graphically, the parabola will touch the x-axis at only one point, which is the vertex of the parabola.
  • If b² - 4ac < 0 (negative): This is where things get interesting! This means we'll have two complex solutions (also known as imaginary solutions). Complex solutions involve the imaginary unit 'i', where i is the square root of -1. This happens because we can't take the square root of a negative number in the realm of real numbers. In this case, the parabola will not intersect the x-axis at all. The discriminant essentially acts as a traffic light for our solutions. A positive discriminant gives us the green light for two real solutions, zero means we have one real solution, and a negative discriminant signals that we're heading into the realm of complex solutions. Understanding the discriminant can save us time and effort because it gives us a sneak peek at what to expect. Before we even plug numbers into the full quadratic formula, we can calculate the discriminant and know whether we're going to be dealing with real or complex solutions. This is a handy trick for checking our work and making sure our final answers make sense. So, the next time you see a quadratic equation, don't forget to give the discriminant a little shout-out – it's the unsung hero of solution-finding!

Applying the Formula to Jonathan's Equation

Okay, guys, let's get back to Jonathan's equation: 0 = 2x² + x + 4. Our mission is to find the solutions, and we're going to use the trusty quadratic formula to do it. The first step is to identify the coefficients 'a', 'b', and 'c'. Remember, the standard form of a quadratic equation is ax² + bx + c = 0.

  • In Jonathan's equation:
    • a = 2 (the coefficient of x²)
    • b = 1 (the coefficient of x)
    • c = 4 (the constant term)

Now that we have our coefficients, we can plug them into the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Substituting the values, we get:

x = (-1 ± √(1² - 4 * 2 * 4)) / (2 * 2)

Let's simplify this step by step. First, we'll deal with the expression under the square root (the discriminant):

1² - 4 * 2 * 4 = 1 - 32 = -31

Notice that the discriminant is negative (-31). This tells us that the solutions will be complex numbers, meaning they will involve the imaginary unit 'i'. Now, let's plug this back into the formula:

x = (-1 ± √(-31)) / 4

Since we can't take the square root of a negative number in the real number system, we introduce the imaginary unit 'i', where i = √(-1). So, we can rewrite √(-31) as √(31 * -1) = √(31) * √(-1) = i√31. Now our equation looks like this:

x = (-1 ± i√31) / 4

This gives us two solutions:

  • x₁ = (-1 + i√31) / 4
  • x₂ = (-1 - i√31) / 4

These are the solutions to Jonathan's quadratic equation. They are complex conjugates, meaning they have the same real part but opposite imaginary parts. This is a common characteristic of quadratic equations with a negative discriminant. So, we've successfully navigated through the equation, identified the coefficients, applied the quadratic formula, and arrived at the complex solutions. Way to go, team!

Identifying the Correct Solution

Great job, everyone! We've successfully found the solutions to Jonathan's equation using the quadratic formula. Now, the final step is to match our solutions with the options provided. Remember, we found two solutions:

  • x₁ = (-1 + i√31) / 4
  • x₂ = (-1 - i√31) / 4

Let's take a look at the options again:

A. (-1 + i√10) / 4 B. (-1 - 3i) / 2 C. (-1 - i√31) / 4 D. (-1 - i√23) / 2

By comparing our solutions with the options, we can see that option C. (-1 - i√31) / 4 perfectly matches one of our solutions, specifically x₂. The other solution, x₁, doesn't match any of the options exactly, but that's okay! Quadratic equations can have two distinct solutions, and we've identified one that is present in the choices. So, the correct answer is undoubtedly option C.

Why Option C is the Perfect Match

Okay, let's really nail down why option C is the winner here. We went through the hard work of using the quadratic formula, and we landed on two solutions: (-1 + i√31) / 4 and (-1 - i√31) / 4. Now, when we look at the answer choices, we're playing a matching game. We're looking for an option that is exactly the same as one of our solutions. Option A, (-1 + i√10) / 4, is close, but no cigar! The imaginary part has a √10, not a √31, so it's out. Option B, (-1 - 3i) / 2, is a totally different beast. The denominator is 2, not 4, and the imaginary part is just a 3i, not an i√31. So, we can confidently reject it. Option D, (-1 - i√23) / 2, is another close-but-no-cigar situation. Again, the denominator is 2, and the imaginary part has a √23, which doesn't match our √31. But then we get to option C, (-1 - i√31) / 4, and it's like the clouds part and the sun shines down! This is a perfect match for one of our solutions. It has the same real part (-1), the same imaginary part (-i√31), and the same denominator (4). It's like finding the missing piece of a puzzle. So, that's why option C is the correct answer. It's not about being almost right; it's about being exactly right. And in the world of math, precision is key! We have to be meticulous in order to ensure the answer is correct.

Conclusion: Mastering Quadratic Equations

Alright, guys! We've done it. We've successfully helped Jonathan find the solutions to the quadratic equation 0 = 2x² + x + 4. We journeyed through the world of quadratic equations, learned about the powerful quadratic formula, and even decoded the secrets of the discriminant. We identified the coefficients, plugged them into the formula, simplified the expression, and arrived at the complex solutions. Then, we carefully matched our solutions with the given options and confidently chose the correct answer.

This whole process highlights the importance of understanding the fundamentals. Knowing what a quadratic equation is, how the quadratic formula works, and what the discriminant tells us is crucial for tackling these problems. It's like having a toolbox full of the right tools for the job. The quadratic formula is a particularly valuable tool because it's reliable and works for any quadratic equation. It's a must-have in your mathematical arsenal. And remember, practice makes perfect! The more you work with quadratic equations, the more comfortable you'll become with the formula and the process. Try solving different equations with varying coefficients and see how the solutions change. Pay attention to the discriminant and how it predicts the nature of the solutions. Don't be afraid to make mistakes – they're part of the learning process. The key is to learn from those mistakes and keep pushing forward.

So, the next time you encounter a quadratic equation, remember Jonathan's equation and the steps we took to solve it. You've got the knowledge, you've got the tools, and you've got the skills to conquer it. Keep practicing, keep exploring, and keep mastering the world of mathematics! You've got this!