Calculate Delta For Quadratic Equation 2x² - 4x + 1 = 0

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Hey guys! Today, we're diving into the fascinating world of quadratic equations and exploring the discriminant, often represented by the Greek letter delta (Δ). Specifically, we're going to tackle the equation 2x² - 4x + 1 = 0. Our mission? To calculate the value of delta using the discriminant formula: Δ = b² - 4ac. Get ready to put on your math hats, because we're about to unravel this quadratic mystery together!

Understanding the Discriminant (Δ)

So, what exactly is the discriminant, and why is it so important? In the realm of quadratic equations, the discriminant, symbolized by delta (Δ), acts as a crucial indicator, revealing the nature and number of roots (or solutions) a quadratic equation possesses. Think of it as a mathematical detective, giving us clues about the equation's behavior without actually solving it completely. The discriminant is nestled within the quadratic formula, which you might remember as the go-to tool for solving quadratic equations of the form ax² + bx + c = 0. The quadratic formula itself is: x = (-b ± √(b² - 4ac)) / 2a. Notice that the expression under the square root, b² - 4ac, is precisely our friend, the discriminant (Δ). Now, let's break down how the value of Δ influences the roots:

  • Δ > 0 (Positive Discriminant): This signals that the quadratic equation has two distinct real roots. Imagine the parabola (the graphical representation of a quadratic equation) intersecting the x-axis at two different points – those points represent the two real roots. It's like finding two separate treasures hidden within the equation!
  • Δ = 0 (Zero Discriminant): A discriminant of zero indicates that the equation has exactly one real root (a repeated root). In this case, the parabola touches the x-axis at just one point, like a perfect bullseye hit. We sometimes call this a double root or a repeated root because it essentially appears twice in the solution.
  • Δ < 0 (Negative Discriminant): When the discriminant turns negative, things get a little more interesting. Here, the quadratic equation has two complex roots. Complex roots involve the imaginary unit 'i' (where i² = -1), and they don't appear on the regular number line. Graphically, this means the parabola doesn't intersect the x-axis at all. It's like the solutions exist in a different mathematical dimension!

In essence, the discriminant is a powerful tool that provides us with valuable insights into the nature of a quadratic equation's solutions even before we embark on the journey of finding those solutions explicitly. By simply calculating Δ, we can predict whether we'll encounter two distinct real roots, a single repeated root, or a pair of complex roots. This understanding can save us time and effort, guiding our problem-solving strategies and deepening our understanding of quadratic equations.

Identifying Coefficients in 2x² - 4x + 1 = 0

Before we can actually calculate the discriminant, we need to pinpoint the coefficients a, b, and c in our given quadratic equation: 2x² - 4x + 1 = 0. Remember, the standard form of a quadratic equation is ax² + bx + c = 0. This form acts as our template, allowing us to easily identify the corresponding values.

Let's break it down step by step:

  • The coefficient 'a': This is the number that multiplies the x² term. In our equation, 2x², the coefficient 'a' is clearly 2. It's like saying we have two times the square of x.
  • The coefficient 'b': This is the number that multiplies the x term. In 2x² - 4x + 1, the term -4x reveals that 'b' is -4. It's crucial to pay attention to the sign! The negative sign is an integral part of the coefficient.
  • The coefficient 'c': This is the constant term, the number that stands alone without any x attached. In our equation, the constant term is +1, so 'c' is simply 1.

So, to recap, we've successfully identified our coefficients:

  • a = 2
  • b = -4
  • c = 1

These values are the building blocks we need to plug into the discriminant formula. Think of them as the ingredients in our quadratic recipe. Now that we have them sorted, we're ready to move on to the exciting part: calculating the actual value of delta!

Calculating Delta (Δ) Using the Formula

Alright, with our coefficients neatly identified (a = 2, b = -4, and c = 1), we're now fully equipped to calculate the discriminant (Δ). Remember the discriminant formula? It's Δ = b² - 4ac. This formula is the key to unlocking the nature of the roots of our quadratic equation.

Let's plug in the values and see what we get:

Δ = (-4)² - 4 * 2 * 1

Now, let's break down the calculation step by step:

  1. (-4)²: Squaring -4 means multiplying it by itself: (-4) * (-4) = 16. Remember, a negative number multiplied by a negative number results in a positive number. So, the first part of our calculation is 16.
  2. 4 * 2 * 1: This is a straightforward multiplication. 4 multiplied by 2 is 8, and 8 multiplied by 1 is still 8. So, the second part of our calculation is 8.
  3. 16 - 8: Now we simply subtract the second result from the first: 16 - 8 = 8.

Therefore, the discriminant (Δ) for the quadratic equation 2x² - 4x + 1 = 0 is 8.

Δ = 8

Woohoo! We've successfully calculated the value of delta. But our journey doesn't end here. We need to interpret what this value tells us about the roots of the equation. A positive discriminant means we have two distinct real roots. This is a crucial piece of information that helps us understand the behavior of the quadratic equation.

Matching the Result to the Alternatives

Okay, we've calculated that the discriminant (Δ) for the equation 2x² - 4x + 1 = 0 is 8. Now, let's see how this aligns with the alternatives provided. Remember, the question gave us these options:

a) 0 b) 2 c) 4 d) 8

It's pretty clear, isn't it? Our calculated value of 8 directly corresponds to option (d). So, the correct answer is:

d) 8

We've successfully navigated the quadratic equation, calculated the discriminant, and matched our result to the given alternatives. Give yourselves a pat on the back, guys! This demonstrates a solid understanding of how to apply the discriminant formula and interpret its result.

Deep Dive into the Nature of Roots

Now that we've nailed the calculation and identified the correct answer, let's take a moment to really understand what our result (Δ = 8) implies about the nature of the roots of the quadratic equation 2x² - 4x + 1 = 0. Remember, the discriminant acts as a window into the soul of the quadratic equation, revealing the characteristics of its solutions without us even having to solve for them explicitly.

In our case, Δ = 8, which is a positive number. This positivity is the key piece of information. When the discriminant is positive (Δ > 0), it tells us that the quadratic equation has two distinct real roots. But what does this actually mean in a more visual and intuitive sense?

Imagine the graph of the quadratic equation, which is a parabola (a U-shaped curve). The roots of the equation are the points where the parabola intersects the x-axis (the horizontal line). When we have two distinct real roots, it means the parabola crosses the x-axis at two different locations. These two points represent the two different real solutions to the equation.

Think of it like this: the x-axis is a treasure map, and the roots are the hidden treasures. A positive discriminant tells us there are two treasure chests buried at different spots along the map. We'll need to dig at both locations to unearth all the riches (the solutions).

Now, let's contrast this with the other possibilities:

  • If Δ = 0 (Zero Discriminant): This would mean the parabola touches the x-axis at only one point. In our treasure map analogy, there's only one treasure chest, and the parabola kisses the x-axis right at that spot.
  • If Δ < 0 (Negative Discriminant): This would mean the parabola doesn't intersect the x-axis at all. Our treasure map is a dud! There are no treasures buried along the x-axis (in the realm of real numbers, at least).

In our specific case, since Δ = 8 is positive, we know for sure that the equation 2x² - 4x + 1 = 0 has two different real solutions. We could go on to use the quadratic formula to actually find those solutions, but the discriminant has already given us valuable information about what to expect. This is the power of the discriminant: it gives us a sneak peek into the solution landscape of a quadratic equation, helping us understand the nature of its roots before we dive into the nitty-gritty calculations.

Conclusion: Mastering the Discriminant

Alright, guys, we've reached the end of our quadratic equation adventure for today! We successfully calculated the discriminant (Δ) for the equation 2x² - 4x + 1 = 0, determined that it's equal to 8, and correctly identified option (d) as the answer. But more importantly, we've explored the fundamental concept of the discriminant and how it acts as a powerful tool for understanding the nature of roots in quadratic equations.

We've learned that the discriminant (Δ = b² - 4ac) is a crucial part of the quadratic formula and that its value tells us whether a quadratic equation has two distinct real roots (Δ > 0), one real root (Δ = 0), or two complex roots (Δ < 0). By simply calculating Δ, we can gain valuable insights into the solutions of the equation without having to go through the entire process of solving it.

In our specific example, the positive discriminant (Δ = 8) revealed that the equation 2x² - 4x + 1 = 0 has two distinct real roots. This understanding allows us to visualize the parabola intersecting the x-axis at two points, representing the two real solutions.

Mastering the discriminant is a key step in becoming a quadratic equation pro. It's like having a secret weapon in your mathematical arsenal! So, keep practicing, keep exploring, and keep unraveling the mysteries of quadratic equations. You've got this!