Solving (X-3)(x+1) + 1 = (x-4)²: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun math problem: solving the equation (X-3)(x+1) + 1 = (x-4)². If you're scratching your head, don't worry! We'll break it down step by step so you can easily understand how to tackle this kind of problem. Let's get started!

Understanding the Equation

Before we jump into solving, let's take a good look at the equation: (X-3)(x+1) + 1 = (x-4)². This is a quadratic equation in disguise. Our goal is to simplify it, get it into a standard form, and then find the value(s) of x that make the equation true. Remember, the standard form of a quadratic equation is ax² + bx + c = 0. We need to get our equation into this format to easily solve for x.

Why is this important? Well, quadratic equations pop up everywhere in math and science. From calculating trajectories in physics to optimizing processes in engineering, understanding how to solve these equations is super useful. Plus, it's a great exercise for your brain!

So, let's roll up our sleeves and start simplifying the equation. First, we'll expand the expressions on both sides. Expanding means multiplying out the terms within the parentheses. This is a crucial step because it helps us get rid of the parentheses and combine like terms, which will eventually lead us to the standard quadratic form. Once we achieve that standard form, we can use various methods like factoring, completing the square, or the quadratic formula to find the solutions for x. Each of these methods has its own advantages and is suited to different types of quadratic equations. The ultimate goal is to isolate x and determine its value or values that satisfy the original equation. Understanding this process is not just about solving this particular problem; it's about equipping yourself with a versatile tool that can be applied to a wide range of mathematical and real-world scenarios.

Step 1: Expand Both Sides

Okay, first, let's expand (X-3)(x+1). We use the FOIL method (First, Outer, Inner, Last):

(X-3)(x+1) = xx + x1 - 3x - 31 = x² + x - 3x - 3 = x² - 2x - 3

Now, let's expand (x-4)²:

(x-4)² = (x-4)(x-4) = xx - 4x - 4*x + 16 = x² - 8x + 16

So, our equation now looks like this:

x² - 2x - 3 + 1 = x² - 8x + 16

Why do we expand? Expanding the equation helps us eliminate parentheses and combine like terms, which is essential for simplifying the equation and moving towards a solution. By multiplying out the terms, we transform the equation into a more manageable form that allows us to isolate the variable x and eventually find its value(s). Expanding is a fundamental algebraic technique that is not only useful in solving equations but also in simplifying complex expressions and understanding the relationships between different terms in an equation.

Step 2: Simplify the Equation

Let's simplify the left side of the equation:

x² - 2x - 3 + 1 = x² - 2x - 2

Now we have:

x² - 2x - 2 = x² - 8x + 16

The next step involves rearranging the equation to bring all terms to one side, aiming to set the equation to zero. This is a crucial step in solving quadratic equations because it allows us to identify the coefficients a, b, and c, which are necessary for applying methods like factoring, completing the square, or using the quadratic formula. In this specific case, we want to eliminate the x² term from both sides to simplify the equation further, as it appears identically on both sides. This simplification will lead us to a linear equation, which is much easier to solve. Understanding how to manipulate and rearrange equations is a fundamental skill in algebra, and it enables us to tackle more complex problems effectively.

Step 3: Eliminate x²

Notice that we have x² on both sides. Let's subtract x² from both sides:

x² - 2x - 2 - x² = x² - 8x + 16 - x²

This simplifies to:

-2x - 2 = -8x + 16

Now we have a linear equation! Linear equations are much simpler to solve than quadratic equations. The key is to isolate the variable x on one side of the equation. We achieve this by performing algebraic operations on both sides, ensuring that we maintain the balance of the equation. The ultimate goal is to get x by itself, which will give us the solution to the equation. Mastering the process of solving linear equations is a fundamental skill in algebra, and it forms the basis for solving more complex mathematical problems. Linear equations are widely used in various fields, including physics, engineering, and economics, making the ability to solve them an invaluable asset.

Step 4: Isolate x

Add 8x to both sides:

-2x - 2 + 8x = -8x + 16 + 8x

This gives us:

6x - 2 = 16

Now, add 2 to both sides:

6x - 2 + 2 = 16 + 2

Which simplifies to:

6x = 18

Finally, divide both sides by 6:

6x / 6 = 18 / 6

So:

x = 3

Step 5: Verify the Solution

It's always a good idea to verify our solution by plugging x = 3 back into the original equation:

(3-3)(3+1) + 1 = (3-4)²

(0)(4) + 1 = (-1)²

0 + 1 = 1

1 = 1

Our solution is correct!

Why is verification important? Plugging the solution back into the original equation ensures that our algebraic manipulations were correct and that we haven't introduced any errors along the way. It's like a final check to confirm that our answer satisfies the initial conditions of the problem. Verification is particularly crucial in complex equations where there's a higher chance of making a mistake. This practice not only validates our solution but also enhances our understanding of the equation and reinforces our problem-solving skills. It's a good habit to develop, as it can save us from submitting incorrect answers and helps build confidence in our mathematical abilities.

Conclusion

So, the solution to the equation (X-3)(x+1) + 1 = (x-4)² is x = 3. I hope this step-by-step guide was helpful! Remember to expand, simplify, isolate, and verify. Keep practicing, and you'll become a math whiz in no time!

Solving equations like this might seem daunting at first, but with a systematic approach and a bit of practice, you can conquer them! Just remember the key steps: expand the expressions, simplify the equation by combining like terms, isolate the variable you're trying to solve for, and always verify your solution by plugging it back into the original equation. Each step is a building block, and as you become more comfortable with these techniques, you'll find that you can tackle increasingly complex problems with confidence. Math is not just about finding the right answer; it's about developing a logical and analytical mindset that can be applied to a wide range of challenges in life.