Find A, B, And C In Quadratic Equation -2x^2 + 4x - 3 = 0

Aug 3, 2025 by Kenji Nakamura 58 views

Decoding the Quadratic Equation -2x^2 + 4x - 3 = 0

Before we dive into identifying the values of a, b, and c in the given quadratic equation, let's take a moment to understand what a quadratic equation is. Guys, in simple terms, a quadratic equation is a polynomial equation of the second degree. This means that the highest power of the variable (usually x) in the equation is 2. The general form of a quadratic equation is expressed as:

ax^2 + bx + c = 0

Where:

a, b, and c are constants, with a not equal to 0.
x represents the variable.

Why is the 'a' not equal to 0' important? If a were 0, the x^2 term would disappear, and the equation would become a linear equation (bx + c = 0) rather than a quadratic one. It's these coefficients—a, b, and c—that dictate the shape and position of the parabola when the quadratic equation is graphed. Understanding the role of these coefficients is super important for solving quadratic equations and understanding their properties.

Identifying a, b, and c is the first step in many methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. The quadratic formula, in particular, relies heavily on the correct identification of these values. It's expressed as:

x = (-b ± √(b^2 - 4ac)) / 2a

This formula is your best friend when you need to find the roots (or solutions) of any quadratic equation, regardless of whether it's easily factorable or not. The part under the square root, b^2 - 4ac, is called the discriminant. The discriminant tells us about the nature of the roots: whether they are real or complex, and whether there are two distinct roots, one repeated root, or no real roots. So, as you can see, a, b, and c are not just arbitrary numbers; they are the key ingredients that determine the behavior and solutions of a quadratic equation. So, let's move on and figure out what those values are in our specific equation!

Alright, let's get down to the nitty-gritty and figure out the values of a, b, and c in the quadratic equation -2x^2 + 4x - 3 = 0. Remember, we're comparing this equation to the general form: ax^2 + bx + c = 0. Guys, it's like matching pieces of a puzzle!

Identifying 'a': The coefficient of the x^2 term is what we're looking for when we want to find 'a.' In our equation, the term with x^2 is -2x^2. So, clearly, the value of a is -2. Notice the negative sign! It's super crucial to include the sign because it changes everything about the equation and its solutions. A positive 'a' means the parabola opens upwards, while a negative 'a' means it opens downwards. This affects the minimum or maximum point of the parabola, and consequently, the nature of the solutions.

Pinpointing 'b': Now, let's move on to 'b.' This is the coefficient of the x term. In our equation, we have +4x. This tells us that b is 4. Easy peasy, right? Just like 'a,' 'b' plays a significant role in determining the parabola's position and orientation. It influences the axis of symmetry of the parabola, which is a vertical line that divides the parabola into two symmetrical halves. The axis of symmetry is given by the formula x = -b / 2a. So, knowing 'b' is essential for finding this crucial line.

Unveiling 'c': Last but not least, we have 'c,' which is the constant term. This is the term that doesn't have any x attached to it. In our equation, we see -3. Therefore, c is -3. Again, pay attention to that negative sign! The constant term 'c' represents the y-intercept of the parabola. It's the point where the parabola crosses the y-axis. So, 'c' gives us a direct visual clue about the graph of the quadratic equation.

Quick Recap:

a = -2
b = 4
c = -3

See how we carefully extracted each value by comparing the given equation to the standard form? This meticulous approach ensures we don't make any mistakes, which is super important when we use these values in further calculations, like the quadratic formula. So, now that we've got our a, b, and c nailed down, we're one step closer to fully understanding this quadratic equation. Let's move on to the next part!

Okay, now that we've successfully identified that a = -2, b = 4, and c = -3 in the quadratic equation -2x^2 + 4x - 3 = 0, it's time to match these values with the correct option provided. This is where attention to detail is key, guys! We need to make sure we pick the option that has all three values spot-on.

Let's quickly recap the options we were given:

A. a = 2, b = 4, c = 3 B. a = 2, b = 4, c = -3 C. a = -2, b = 4, c = 3 D. a = -2, b = 4, c = -3

Now, let's compare our values with each option:

Option A: This option has a = 2, but we know a is actually -2. So, this one's a no-go.
Option B: This option also has a = 2, which is incorrect. So, we can eliminate this one too.
Option C: Here, a = -2, which matches our finding. However, c = 3, but we know c should be -3. So, close, but not quite!
Option D: This option has a = -2, b = 4, and c = -3. Bingo! This perfectly matches the values we identified.

Therefore, the correct answer is Option D: a = -2, b = 4, c = -3.

See how important it is to be meticulous and check each value? A small mistake in identifying even one of these values can lead to the wrong answer, especially when you're dealing with more complex problems that build upon these basics. So, always double-check and make sure everything lines up perfectly. Now that we've nailed this, let's think about why this is important and how it fits into the bigger picture of solving quadratic equations.

Alright, guys, we've successfully identified the values of a, b, and c in the given quadratic equation and matched them to the correct option. But you might be thinking, “Okay, cool, we did it… but why does this even matter?” That's a great question! Understanding why this process is important can make learning math way more engaging and less like just memorizing steps.

The ability to correctly identify a, b, and c is a foundational skill in solving quadratic equations. It's like knowing your ABCs before you can write a sentence. These coefficients are the building blocks for so many different methods and concepts related to quadratics. Think of it this way: a, b, and c are the key ingredients in a recipe, and quadratic equations are the delicious dishes we want to cook up!

Here's why it's super important:

The Quadratic Formula: We talked about this earlier, but it's worth emphasizing. The quadratic formula is a universal tool for solving quadratic equations, and it directly uses the values of a, b, and c. If you mess up identifying these, your final answer will be wrong. It's like adding the wrong amount of salt to a dish – it can throw everything off!
Factoring: While not all quadratic equations can be easily factored, for those that can, knowing a, b, and c helps you break down the equation into simpler terms. Factoring is a neat trick for finding the roots quickly, and it relies on understanding the relationships between these coefficients.
Completing the Square: This is another method for solving quadratic equations, and guess what? It also needs you to know a, b, and c. Completing the square transforms the equation into a perfect square trinomial, making it easier to solve. It's a bit like reshaping a puzzle to fit the pieces together perfectly.
Graphing Quadratic Equations: Remember that parabolas are the graphs of quadratic equations. The values of a, b, and c tell you a lot about the parabola's shape, position, and direction. 'a' tells you if it opens upwards or downwards, 'c' gives you the y-intercept, and '-b/2a' helps you find the axis of symmetry. So, knowing these values is like having a map to navigate the world of parabolas!
Understanding the Discriminant: The discriminant (b^2 - 4ac) is a part of the quadratic formula, and it tells you about the nature of the roots of the equation. It tells you whether you have two real roots, one real root, or no real roots (complex roots). So, by correctly identifying a, b, and c, you can predict what kind of solutions to expect.

In essence, mastering the identification of a, b, and c is like unlocking a secret code to the world of quadratic equations. It empowers you to solve problems, understand concepts, and build a solid foundation for more advanced math topics. So, keep practicing, stay curious, and remember that every step you take in understanding these basics is a step towards becoming a math whiz!

So, guys, we've reached the end of our journey in dissecting the quadratic equation -2x^2 + 4x - 3 = 0. We started by understanding the general form of a quadratic equation, then meticulously identified the values of a, b, and c, matched them to the correct option, and finally, we explored why this skill is so crucial in the broader context of quadratic equations. Phew! That's quite a bit, but hopefully, it's all crystal clear now.

Let's wrap things up with some key takeaways to really solidify your understanding:

The General Form is Your Friend: Always remember the general form of a quadratic equation: ax^2 + bx + c = 0. This is your reference point for identifying the coefficients.
Pay Attention to Signs: The signs (+ or -) of a, b, and c are super important. Don't overlook them! A negative sign can completely change the nature of the equation and its solutions.
Each Coefficient Has a Role: a determines the direction of the parabola's opening, b influences the axis of symmetry, and c gives you the y-intercept. Knowing this helps you visualize the equation and its graph.
Practice Makes Perfect: Like any skill, identifying a, b, and c becomes easier with practice. Work through different examples, and you'll become a pro in no time!
It's a Foundation: This skill is the bedrock for solving quadratic equations using various methods like the quadratic formula, factoring, and completing the square. Master this, and you'll be well-equipped for more advanced math topics.

Identifying the values of a, b, and c in a quadratic equation might seem like a small step, but it's a giant leap in your mathematical journey. It's about building a strong foundation, developing problem-solving skills, and understanding the underlying concepts. Math isn't just about memorizing formulas; it's about understanding how things work and applying that knowledge to solve problems.

So, keep exploring, keep questioning, and keep practicing. You've got this! And who knows, maybe you'll even start to see quadratic equations in the world around you – in the curve of a bridge, the trajectory of a ball, or the shape of a satellite dish. Math is everywhere, guys, and understanding it opens up a whole new way of seeing the world. Keep up the awesome work!