Solving Quadratic Equations: Step-by-Step Guide

Oct 29, 2025 by Andrew McMorgan 48 views

Hey Plastik Magazine readers! Today, we're diving deep into the world of solving quadratic equations. Don't worry, it's not as scary as it sounds! We'll walk through a specific problem, breaking it down into easy-to-understand steps. This isn’t just about getting the answer; it's about understanding the process so you can tackle similar problems with confidence. Get ready to flex those math muscles, because we're about to solve this equation: $\frac{x^2-x-6}{x^2}=\frac{x-6}{2 x}+\frac{2 x+12}{x}$ .

The Problem: Unveiling the Quadratic Equation

So, the main question is: how do we solve the equation $\frac{x^2-x-6}{x^2}=\frac{x-6}{2 x}+\frac{2 x+12}{x}$ ? This equation looks a little messy, right? It involves fractions with 'x' in the denominator. Our goal is to simplify this equation and find the value(s) of 'x' that make it true. This is where understanding the fundamentals of algebra is super important. We will use the fundamentals to move from the given expression to the final result, and along the way we will highlight key steps.

First, let's identify the type of equation we are dealing with. Although the equation initially looks complex due to the fractions, our aim is to transform it into a more manageable form, specifically a quadratic equation. A quadratic equation, in its standard form, is written as $ax^2 + bx + c = 0$ , where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. This is the ultimate destination, the format we aim to achieve after simplifying the original equation. The process involves eliminating the fractions, which is the initial obstacle. This is generally done by multiplying the entire equation by a common denominator.

To make sure we're on the right track, let's make sure our math vocabulary is up to par. Terms like “quadratic equation,” “variable,” and “constant” will be used throughout the process. A variable is a symbol (usually a letter, like 'x') that represents an unknown value. A constant, on the other hand, is a fixed number. Understanding these terms will help you feel more comfortable as we proceed. The ultimate goal is to isolate the variable on one side of the equation to determine its value. This is the crux of solving the equation; isolating x will allow us to find the roots of the equation.

Step 1: Finding the Least Common Denominator (LCD)

Alright, guys, before we get started with the actual solving, let's talk about the first step: finding the Least Common Denominator (LCD). The LCD is the smallest expression that all the denominators in our equation can divide into evenly. Think of it as the magic number that will help us get rid of those pesky fractions. In our equation, the denominators are $x^2$ , $2x$ , and $x$ . The LCD is the product of the highest powers of all the factors present in the denominators. Thus, to find the LCD, we need to consider each denominator. Here’s how we break it down:

$x^2$ is already factored.
$2x$ has factors 2 and x.
$x$ is already factored.

Combining these, we get an LCD of $2x^2$ . This is because $2x^2$ is the smallest expression divisible by each of our original denominators ( $x^2$ , $2x$ , and $x$ ). It includes all the factors necessary to eliminate the fractions, setting us up for a smoother solution. Remember that finding the LCD is crucial because it ensures that when we multiply the equation, we’re doing it efficiently. A well-chosen LCD simplifies the calculations.

Step 2: Multiplying by the LCD

Now for the fun part! We multiply each term in our equation by the LCD, which is $2x^2$ . This is like giving each term a fair shake, making sure we apply the same operation across the board. The whole point here is to eliminate the fractions. This might seem a bit tedious, but trust me, it’s worth it. When multiplying by the LCD, we are essentially clearing the denominators.

Let’s go through this step by step. We have the equation $\frac{x^2-x-6}{x^2}=\frac{x-6}{2 x}+\frac{2 x+12}{x}$ . Multiplying each term by $2x^2$ , we get:

$2x^2 * \frac{x^2-x-6}{x^2} = 2x^2 * \frac{x-6}{2x} + 2x^2 * \frac{2x+12}{x}$

Let's break down each multiplication to show how the fractions disappear:

For the first term, the $x^2$ in the numerator and denominator cancel out, leaving us with $2(x^2 - x - 6)$ .
For the second term, the $2x$ in the denominator cancels with a $2x$ in the $2x^2$ , leaving us with $x(x - 6)$ .
For the third term, the $x$ in the denominator cancels with an $x$ in the $2x^2$ , leaving us with $2x(2x + 12)$ .

This leaves us with the equation: $2(x^2 - x - 6) = x(x - 6) + 2x(2x + 12)$ .

Step 3: Simplifying and Expanding

Next, we need to simplify and expand the equation. We’re getting closer to that standard quadratic form! This involves multiplying out the terms and collecting like terms. Expanding and simplifying is more about algebra manipulation to get the equation closer to our standard form. The goal is to consolidate terms, combining similar elements into a simplified format. This prepares the equation for the final steps, where we will solve for x.

Expanding each term gives us: $2x^2 - 2x - 12 = x^2 - 6x + 4x^2 + 24x$ . Now, let's combine like terms on the right side of the equation. We have $x^2$ and $4x^2$ which add up to $5x^2$ . We also have $-6x$ and $24x$ which sum to $18x$ . This simplifies our equation to: $2x^2 - 2x - 12 = 5x^2 + 18x$ . Now, we want to bring all the terms to one side of the equation. Subtracting $2x^2$ , adding $2x$ and adding $12$ to both sides, the equation becomes $0 = 3x^2 + 20x + 12$ .

Step 4: Solving the Quadratic Equation

Now, let's rearrange our equation so it looks like the standard quadratic form $ax^2 + bx + c = 0$ . After simplifying, the equation becomes $3x^2 + 20x + 12 = 0$ . We've successfully transformed our original equation into a standard quadratic form, which makes it easier to solve for 'x'. This is a critical step because it allows us to apply standard methods for solving quadratic equations.

We now look at the provided options to determine which one is correct. We can see that the equation we have matches option C: $3x^2 + 20x + 12 = 0$ . Therefore, we can confidently mark C as the correct answer. We've solved for the value of x by transforming the original equation into a solvable form, simplifying the process and making it accessible. This careful, step-by-step approach not only helps us find the right answer but also equips us with the tools to solve a variety of quadratic equations.

Conclusion: Mastering Quadratic Equations

And there you have it, guys! We've successfully solved the quadratic equation. Remember, breaking down a problem into smaller, manageable steps is key. By finding the LCD, multiplying to clear fractions, simplifying, and then rearranging into the standard form, we were able to find the correct answer. This entire process highlights the importance of mastering fundamental algebraic techniques.

Summary of the Steps

Let’s recap what we did:

Find the LCD: Identify the least common denominator to eliminate fractions.
Multiply by the LCD: Multiply each term in the equation by the LCD.
Simplify and Expand: Simplify and expand the equation.
Solve the Quadratic Equation: Rearrange the equation and find the values of x.

Keep practicing these steps, and you'll become a pro at solving quadratic equations in no time! Keep exploring and challenging yourselves with new problems.

Thanks for tuning in, and until next time, happy solving!