Solving Quadratic Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of quadratic equations! Finding the roots of these equations might seem intimidating at first, but trust me, with a little practice and the right approach, you'll be solving them like a pro. This guide will walk you through solving the equations: a) x² + 10x + 24 = 0, b) x² - 225 = 0, and c) 1/7 4x = 0. We'll break down each equation step-by-step, explaining the methods and concepts involved. So, grab your pencils, get comfy, and let's get started!

Understanding Quadratic Equations and Their Roots

Before we start solving, let's make sure we're on the same page. Quadratic equations are equations of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The roots of a quadratic equation are the values of 'x' that satisfy the equation, meaning the values of 'x' that make the equation true. These roots represent the points where the parabola (the graph of a quadratic equation) intersects the x-axis. There are several methods to find these roots, including factoring, using the quadratic formula, and completing the square. The best method to use depends on the specific equation and your preference. Remember that a quadratic equation can have two distinct real roots, one repeated real root, or two complex roots. Let's start with a) x² + 10x + 24 = 0. This is a standard quadratic equation, and we can solve it by factoring. Factoring is the process of breaking down a quadratic expression into the product of two binomials. The goal is to find two numbers that multiply to give the constant term (24 in this case) and add up to the coefficient of the x term (10 in this case). Always remember the basic definition to solve any quadratic equation. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and x is the variable we need to solve for. The roots, also known as solutions, are the values of x that make the equation true, these values are where the graph of the equation crosses the x-axis. Now, let's get our hands dirty and start solving these equations!

Solving x² + 10x + 24 = 0: Factoring Method

Okay, let's tackle the first equation, x² + 10x + 24 = 0. We'll use the factoring method here. The factoring method is a straightforward approach that involves breaking down the quadratic expression into two binomials. To factor this equation, we need to find two numbers that multiply to give us 24 (the constant term) and add up to 10 (the coefficient of the x term). After some thought, we find that the numbers 6 and 4 fit the bill because 6 * 4 = 24 and 6 + 4 = 10. So, we can rewrite the equation as (x + 6)(x + 4) = 0. For this equation to be true, either (x + 6) = 0 or (x + 4) = 0. Solving these two simple linear equations, we get x = -6 and x = -4. Therefore, the roots of the equation x² + 10x + 24 = 0 are x = -6 and x = -4. These are the values of 'x' that satisfy the original equation. If you substitute -6 or -4 into the equation, you'll find that the equation equals zero, confirming that these are indeed the roots. The beauty of factoring is that it provides a direct way to find the roots if the quadratic expression can be easily factored. The factoring method is great because it gives you a clean way to find the solutions if the equation is factorable. Remember to always double-check your solutions by plugging them back into the original equation to ensure they are correct. Now that we have solved this, let's proceed to the next equation.

Solving x² - 225 = 0: Difference of Squares

Alright, let's move on to the second equation: x² - 225 = 0. This equation is a bit different from the first one. It's a special type of quadratic equation known as a difference of squares. The difference of squares is an algebraic identity that states a² - b² = (a + b)(a - b). In our equation, x² - 225 = 0, we can recognize that 225 is a perfect square (15² = 225). This allows us to apply the difference of squares identity. We can rewrite the equation as x² - 15² = 0. Using the difference of squares identity, we get (x + 15)(x - 15) = 0. This means either (x + 15) = 0 or (x - 15) = 0. Solving these, we find x = -15 and x = 15. So, the roots of the equation x² - 225 = 0 are x = -15 and x = 15. The difference of squares is a very useful technique to solve equations of this form quickly. It's a shortcut that simplifies the process, making it much faster to find the roots. Always be on the lookout for perfect squares to spot a difference of squares opportunity. This method is particularly useful because it allows us to bypass more complex methods, saving time and effort. Also, remember to always verify your solutions! Now, let us tackle the final equation.

Solving 1/7 4x = 0: Linear Equation

Okay, now let's solve the last equation, which is actually a linear equation and not a quadratic one, despite the initial formatting. The equation is 1/7 4x = 0. To solve this, we want to isolate 'x'. First, multiply both sides of the equation by 7 to get rid of the fraction: 7 * (1/7 4x) = 7 * 0, simplifying to 4x = 0. Now, divide both sides by 4: 4x / 4 = 0 / 4, which gives us x = 0. Therefore, the solution to the equation 1/7 4x = 0 is x = 0. This kind of equation is much simpler than quadratic equations because it involves only linear terms. It’s a great example of how important it is to recognize the type of equation you're dealing with, as different types require different approaches. In this case, it was a simple linear equation, not a quadratic one. Solving linear equations is fundamental in algebra. This equation showcases a simple application of algebraic principles. Understanding and quickly solving these types of equations is essential for building a solid foundation in mathematics. It's really all about isolating the variable 'x' on one side of the equation. Always ensure you are following the correct steps to avoid making arithmetic errors. We have solved all the equations given to us; congrats!

Conclusion: Mastering Quadratic Equations

Great job, guys! We've successfully solved all the given equations. We've used different methods such as factoring, recognizing the difference of squares, and solving a linear equation. Remember that practice is key to mastering these concepts. The more you solve these types of equations, the more comfortable you'll become, and the faster you'll be able to solve them. Don't be afraid to try different approaches or to check your answers. Always double-check your work, and don't be discouraged if you don't get it right away. The main point is to understand the concepts and the steps involved. By understanding these core concepts, you'll be well-equipped to tackle more complex problems in the future. Keep practicing, keep learning, and keep asking questions. If you get stuck, don't worry, just revisit the steps or ask for help. Remember, math is a journey, and with each equation you solve, you're gaining more confidence and skills. Keep practicing these methods, and soon you'll be solving quadratic equations like a boss! I hope this step-by-step guide has been helpful. Keep up the amazing work, and keep exploring the amazing world of mathematics! Don’t hesitate to return to this guide whenever you feel a need to refresh your memory. Keep practicing and keep up the great work!