Solving Quadratic Equations: Step-by-Step Guide

Hey math enthusiasts! Let's dive into the world of quadratic equations. We're going to break down how to solve them step-by-step, making it super easy to understand. In this article, we'll focus on solving the equation x² - 2x - 30 = 5, but the principles we cover will apply to all quadratic equations. So, grab your pencils, and let's get started!

Understanding Quadratic Equations

First things first, what exactly is a quadratic equation? Simply put, it's an equation that can be written in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The 'x' is our variable, and we're trying to find its value(s) that make the equation true. These equations are called 'quadratic' because the highest power of the variable (in this case, 'x') is 2. The standard form is important because it sets the stage for how we solve these equations. Think of it like a roadmap; it shows us the direction to head in to find the solutions. The values of a, b, and c are critical, as they dictate the behavior of the equation and its solutions. For example, if 'a' is positive, the parabola (the shape of the graph of the equation) opens upwards, and if 'a' is negative, it opens downwards. This helps in understanding the nature of the solutions we are seeking. The solutions, also known as roots or zeros, represent the points where the parabola intersects the x-axis. A quadratic equation can have two real solutions, one real solution (a repeated root), or two complex solutions. The nature of the solutions depends on the discriminant (b² - 4ac), which we will explore later. The goal is always to find the value or values of x that satisfies the equation. It's a fundamental concept in algebra, and mastering it unlocks a deeper understanding of mathematical relationships.

Now, back to our equation x² - 2x - 30 = 5. This looks a little different from our standard form, right? Not to worry, the first step is always to get it into that standard form, which is ax² + bx + c = 0. This means we need to manipulate the equation so that everything is on one side and zero is on the other. It's like balancing a scale; we want to make sure both sides are equal. It is really important to know, that the quadratic equation has a lot of real-world applications. It's used in physics to model projectile motion, in engineering to design structures, and in finance to calculate investments. So, by solving these equations, you're not just doing math; you're gaining the tools to understand and model various aspects of the world around us. In addition, the ability to solve quadratic equations is a key building block for more advanced mathematical concepts, like calculus and differential equations. Getting this down now will serve you well in the future. Quadratic equations are essential for understanding how the world works.

Step-by-Step Solution

Alright, let's get down to business and solve x² - 2x - 30 = 5. Here's a step-by-step guide to help you out:

Step 1: Rewrite the Equation in Standard Form

Our first move is to transform the equation into the standard form ax² + bx + c = 0. To do this, we'll subtract 5 from both sides of the equation. This gives us:

x² - 2x - 35 = 0

See? It's all about making sure everything is on one side, and zero is on the other. This form makes it easy to identify our a, b, and c values, which are critical for the next steps. It is also important to remember that we need to keep the equation balanced during this operation. Whatever we do on one side, we must also do on the other side. This ensures that the equality remains true throughout the process. The standard form is the key to applying various solution methods, such as factoring, completing the square, or using the quadratic formula. Each method is suited to different types of quadratic equations, and understanding the standard form allows us to choose the most appropriate method. Plus, it just makes the equation cleaner and easier to work with! The standard form also clearly shows us the coefficients we will use to find the roots of the equation. Always get it into this form before you proceed.

Step 2: Identify a, b, and c

Now that we have the equation in standard form x² - 2x - 35 = 0, let's identify the values of a, b, and c. These values are the coefficients of the quadratic equation. In this case:

• a = 1 (the coefficient of x²)
• b = -2 (the coefficient of x)
• c = -35 (the constant term)

Knowing these values is really important, as they dictate the method we choose to solve the equation. The values also become crucial when we apply the quadratic formula. Identifying a, b, and c is like knowing the ingredients before you start cooking a meal. It provides a quick way to plug them into the quadratic formula to solve for x. Remember that even though the x² term looks like it has no coefficient, there is an invisible 1 in front of it. Also, be careful with the signs! Make sure you take into account the negative signs when identifying b and c. The coefficients determine the parabola's shape, position, and orientation. Understanding these coefficients gives us a deeper insight into the equation. The process might seem simple, but this step is fundamental to correctly applying the solution methods. The next steps depend on knowing your a, b and c values!

Step 3: Choose a Solution Method

We have a few methods available at our disposal to solve quadratic equations. Here are the most common ones:

• Factoring: This involves rewriting the quadratic expression as a product of two binomials. This method is fast if the equation is factorable but may not work for all equations.
• Completing the Square: This method involves manipulating the equation to create a perfect square trinomial. It works for all quadratic equations but can be more time-consuming.
• Quadratic Formula: This formula provides a direct solution for x using the values of a, b, and c. It works for all quadratic equations and is often the most reliable method.

For our equation x² - 2x - 35 = 0, let's try factoring first. It's often the quickest way if the equation is easily factorable.

Choosing the right method can really save you some time and effort. Factoring is like finding a shortcut. If you can factor the equation, you can get the solutions quickly. If factoring doesn't work, don't worry! Completing the square is another option. And if all else fails, the quadratic formula will always get you to the solution. The factoring method is not always applicable, while the quadratic formula can always be used, regardless of the equation's structure. Therefore, the choice of method depends on the equation's specific characteristics and your personal preference. Remember, practice is key! The more you work with quadratic equations, the more familiar you will become with each method and when to use them.

Step 4: Solve by Factoring

Okay, let's try to factor the equation x² - 2x - 35 = 0. We need to find two numbers that multiply to give us -35 (the value of 'c') and add up to -2 (the value of 'b').

After some thought, we find that -7 and 5 fit the bill because (-7) * (5) = -35 and (-7) + (5) = -2. So, we can rewrite the equation as:

(x - 7)(x + 5) = 0

To find the solutions for x, we set each factor equal to zero:

• x - 7 = 0 => x = 7
• x + 5 = 0 => x = -5

So, the solutions to our equation are x = 7 and x = -5.

Factoring can seem like a puzzle, but it gets easier with practice. The ability to quickly spot the right factors is the key. The factors represent the x-intercepts of the parabola. When you can factor, you're essentially finding the points where the parabola crosses the x-axis. This is the same as the roots of the equation. Keep in mind that not all quadratic equations can be easily factored, which is why other methods, such as the quadratic formula, are important to know. Factoring is all about splitting the quadratic expression into two linear factors. This way, we simplify the equation into something that's much easier to solve. The factors that you discover are the solutions to your equation!

Step 5: Verification of the Solution

Always verify your solutions to ensure that your values are correct. To verify our solutions, plug them back into the original equation x² - 2x - 30 = 5 and confirm that it equals 5.

Let's start with x = 7:

(7)² - 2*(7) - 30 = 49 - 14 - 30 = 5

Now, let's try x = -5:

(-5)² - 2*(-5) - 30 = 25 + 10 - 30 = 5

Both solutions satisfy the original equation, so we have found the correct answers!

Checking your work is super important. It is always a good idea to make sure that the solutions you get are correct. This can prevent errors and help you catch any mistakes you might have made in the steps. Plugging your answers back into the original equation is like a safety net; it ensures that your answers are correct. Verifying the results is also essential for building confidence in your problem-solving skills. It is especially useful in exams, where you want to be sure you are right. Always take the time to check your answers! Keep in mind that you're not just solving an equation; you're building a deeper understanding of the relationships between variables and numbers. Always make sure that you are double-checking your math to make sure you have the right answers! This will also help you to catch any mistakes you've made. Always double check your solutions to ensure that they are correct.

Conclusion

And there you have it! We've successfully solved the quadratic equation x² - 2x - 30 = 5. We converted the equation into standard form, identified a, b, and c, chose to factor the equation, and found our solutions. Remember, practice makes perfect. Keep working through problems, and you'll become a pro at solving quadratic equations. This is a fundamental skill in math that will open doors to more advanced concepts. Now go out there and conquer those equations, guys!