Solving The Quadratic Equation: (1/4)x^2 = -(1/2)x + 2

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Hey guys! Today, let's dive into solving a quadratic equation. We're going to tackle the equation (1/4)x^2 = -(1/2)x + 2. It might look a bit intimidating at first, but don't worry, we'll break it down step by step. Understanding how to solve quadratic equations is super important in mathematics, as they pop up everywhere from physics problems to engineering calculations. So, grab your thinking caps, and let's get started!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's quickly recap what quadratic equations are all about. A quadratic equation is essentially a polynomial equation of the second degree. That 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 isn't zero (otherwise, it wouldn't be a quadratic equation!).

The solutions to a quadratic equation, also known as roots or zeros, are the values of 'x' that make the equation true. A quadratic equation can have up to two real solutions, one real solution (a repeated root), or two complex solutions. There are several methods to find these solutions, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and is suitable for different types of quadratic equations. We'll be using a combination of algebraic manipulation and factoring in our case today. But remember guys, practice makes perfect! The more you work with quadratic equations, the more comfortable you'll become with identifying the best method for solving them.

Transforming the Equation

Okay, so let's get back to our equation: (1/4)x^2 = -(1/2)x + 2. The first thing we want to do is get it into that standard quadratic form ax² + bx + c = 0. This makes it much easier to work with. To do this, we need to move all the terms to one side of the equation.

Step 1: Eliminate Fractions

Fractions can be a bit messy to deal with, so let's get rid of them first. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which in this case is 4. This gives us:

4 * (1/4)x^2 = 4 * [-(1/2)x + 2]

This simplifies to:

x^2 = -2x + 8

See? Much cleaner already! This step is crucial because it transforms the equation into a more manageable form, eliminating the fractions that often cause confusion. By multiplying both sides by the LCM, we ensure that the equation remains balanced while simplifying the coefficients. This makes the subsequent steps of rearranging and solving the equation significantly easier. It's like clearing the clutter before you start a project – it sets you up for success!

Step 2: Rearrange into Standard Form

Now, let's move all the terms to the left side of the equation to get it into the standard form ax² + bx + c = 0. We can do this by adding 2x and subtracting 8 from both sides:

x^2 + 2x - 8 = 0

Great! Now our equation is in the familiar quadratic form. This is a key step because many solution methods, like factoring and the quadratic formula, rely on having the equation in this form. By rearranging the terms, we've created a clear structure that allows us to easily identify the coefficients a, b, and c, which are essential for applying these methods. Think of it as organizing your tools before starting a task – having everything in its place makes the job much smoother.

Solving by Factoring

Now that we have our equation in standard form, x^2 + 2x - 8 = 0, we can try to solve it by factoring. Factoring involves breaking down the quadratic expression into the product of two binomials. This method is particularly efficient when the quadratic expression can be easily factored. But don't worry guys, even if factoring isn't your strong suit, there are other methods we can use later!

Step 1: Find Two Numbers

We need to find two numbers that multiply to -8 (the constant term) and add up to 2 (the coefficient of the x term). Let's think about the factors of -8:

  • -1 and 8
  • -2 and 4
  • -4 and 2
  • -8 and 1

Out of these pairs, -2 and 4 add up to 2. Bingo! Finding these numbers is the crucial step in factoring. It's like solving a little puzzle within the larger equation. These numbers will become the constant terms in our two binomial factors. The process of identifying these numbers often involves some trial and error, but with practice, you'll start to recognize patterns and become quicker at it.

Step 2: Factor the Quadratic

Now we can rewrite the quadratic expression using the numbers we found:

x^2 + 2x - 8 = (x - 2)(x + 4)

So, our equation becomes:

(x - 2)(x + 4) = 0

This step transforms the quadratic equation from a sum of terms into a product of factors, which is the heart of the factoring method. Each factor represents a potential solution, and the next step will reveal how these factors lead us to the roots of the equation. This factorization showcases the power of rewriting expressions in different forms to simplify the solving process.

Step 3: Solve for x

For the product of two factors to be zero, at least one of them must be zero. So, we set each factor equal to zero and solve for x:

x - 2 = 0 or x + 4 = 0

Solving these simple equations gives us:

x = 2 or x = -4

These are the solutions to our quadratic equation! This is the moment of truth where the factors reveal the solutions. By setting each factor to zero, we're essentially finding the values of x that make the entire equation true. These solutions are the points where the parabola represented by the quadratic equation intersects the x-axis. It's like finding the hidden keys that unlock the equation's secrets.

The Solutions

Therefore, the solutions to the equation (1/4)x^2 = -(1/2)x + 2 are x = -4 and x = 2. So, the correct answer is A. -4 and 2.

We did it! We successfully solved the quadratic equation by transforming it into standard form and then using factoring. Remember, guys, there are often multiple ways to solve a problem in math, so if factoring doesn't click for you, there are other methods like the quadratic formula that you can explore. The most important thing is to understand the underlying concepts and practice, practice, practice!

Alternative Methods (Brief Overview)

While we solved this equation by factoring, let's briefly touch upon other methods you can use to solve quadratic equations. Knowing different methods gives you flexibility and a deeper understanding of the topic.

  • Quadratic Formula: The quadratic formula is a universal tool that can solve any quadratic equation. It's especially useful when factoring is difficult or impossible. The formula is: x = (-b ± √(b² - 4ac)) / 2a. Just plug in the values of a, b, and c from the standard form of the equation, and you'll get the solutions.
  • Completing the Square: Completing the square is another method that involves transforming the quadratic equation into a perfect square trinomial. It's a bit more involved than factoring, but it's a powerful technique that can also be used to derive the quadratic formula.

Conclusion

Solving quadratic equations is a fundamental skill in mathematics, and we've seen how to tackle one using factoring. Remember the key steps: transform the equation into standard form, find the factors (if possible), and set each factor equal to zero to find the solutions. And don't forget, there are other methods like the quadratic formula and completing the square that you can use as well. Keep practicing, and you'll become a quadratic equation-solving pro in no time! Keep up the great work guys!