Solving (x-5)^2 = 1: A Step-by-Step Guide

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Hey guys! Today, we're diving into a fun little math problem that involves solving the equation $(x-5)^2 = 1$. This type of problem is super common in algebra, and mastering it will definitely boost your math skills. We'll break it down step-by-step, making sure everyone understands the process. So, let's get started!

Understanding the Problem

Before we jump into solving, let's take a moment to understand what the equation $(x-5)^2 = 1$ actually means. In essence, we are looking for values of x that, when we subtract 5 from them and then square the result, we get 1. This equation is a quadratic equation in disguise, and there are a couple of ways we can approach solving it. The key here is to remember that squaring a number means multiplying it by itself, and there are often two numbers (a positive and a negative) that will give you the same square. For example, both 1 and -1, when squared, result in 1. This is a crucial concept that we’ll use to find our solutions.

Breaking Down the Square

The expression $(x-5)^2$ means $(x-5)$ multiplied by itself. So, we can rewrite the equation as $(x-5)(x-5) = 1$. Now, you might be tempted to expand this using the FOIL (First, Outer, Inner, Last) method, but there’s a more elegant and quicker way to solve this particular equation. The reason we can avoid expanding is the presence of the perfect square and the constant on the other side of the equation. Recognizing this structure allows us to use the square root property, which simplifies the solution process significantly. This method not only saves time but also reduces the chances of making algebraic errors along the way. Think of it as a shortcut that experienced math solvers often use!

The Square Root Property

The square root property is our best friend here. It tells us that if $a^2 = b$, then $a = \sqrt{b}$ or $a = -\sqrt{b}$. In simpler terms, if something squared equals a number, then that something can be either the positive or the negative square root of the number. This is because both the positive and negative versions of a number, when squared, will give a positive result. Applying this to our equation, we see that $(x-5)^2 = 1$, so $(x-5)$ can be either the square root of 1 or the negative square root of 1. This gives us two separate equations to solve, each leading to a potential solution for x.

Solving the Equation

Okay, let's get our hands dirty and solve this thing! We'll use the square root property we just discussed.

Step 1: Apply the Square Root Property

As we said, $(x-5)^2 = 1$ implies that $x-5$ can be either $\sqrt{1}$ or $-\sqrt{1}$. Since $\sqrt{1} = 1$, we have two equations:

1. $x - 5 = 1$
2. $x - 5 = -1$

See how we've split our original equation into two simpler equations? This is the power of the square root property! We've transformed a quadratic-like equation into two linear equations, which are much easier to solve.

Step 2: Solve for x in Each Equation

Now, let's solve each equation separately. This involves isolating x on one side of the equation.

• Equation 1: $x - 5 = 1$ To isolate x, we add 5 to both sides of the equation: $x - 5 + 5 = 1 + 5$ $x = 6$

  So, one solution is $x = 6$.

• Equation 2: $x - 5 = -1$ Again, we add 5 to both sides: $x - 5 + 5 = -1 + 5$ $x = 4$

  And our second solution is $x = 4$.

Step 3: Check Your Solutions

It's always a good idea to check your solutions to make sure they're correct. We'll plug each value of x back into the original equation and see if it holds true.

• Check x = 6: $(6 - 5)^2 = 1^2 = 1$. This is correct!

• Check x = 4: $(4 - 5)^2 = (-1)^2 = 1$. This is also correct!

Both solutions check out, so we can be confident in our answer.

Analyzing the Answer Choices

Now that we've solved the equation and found our solutions ($x = 4$ and $x = 6$), let's look at the answer choices provided and see which one matches our findings.

a. $x = 4$ or $x = 6$ b. $x = -6$ or $x = -4$ c. $x = -6$ or $x = 4$ d. $x = -4$ or $x = 6$

It's clear that option a is the correct answer, as it includes both of our solutions: $x = 4$ and $x = 6$. The other options include incorrect values or the wrong signs.

Why Other Options Are Incorrect

It's helpful to understand why the other options are incorrect. This not only reinforces the correct solution but also helps you avoid similar mistakes in the future. Let's break down why options b, c, and d are not the correct answers:

• Option b: $x = -6$ or $x = -4$ This option suggests negative values for x. If we plug these values into the original equation, we can quickly see they don't work:

  • For $x = -6$: $(-6 - 5)^2 = (-11)^2 = 121$, which is not equal to 1.
  • For $x = -4$: $(-4 - 5)^2 = (-9)^2 = 81$, which is also not equal to 1. These negative values lead to much larger results when squared, so they cannot be solutions.

• Option c: $x = -6$ or $x = 4$ This option mixes a negative value ($x = -6$) with one of the correct solutions ($x = 4$). We already showed that $x = -6$ doesn't work, so this option is incorrect.

• Option d: $x = -4$ or $x = 6$ Similar to option c, this option includes a negative value ($x = -4$) along with one of the correct solutions ($x = 6$). Again, $x = -4$ doesn't satisfy the original equation.

Understanding why these options fail is a great way to solidify your understanding of the solution process and the importance of checking your answers.

Alternative Method: Expanding and Factoring

While we used the square root property to solve this equation quickly, there's another method you might be familiar with: expanding the equation and then factoring. Let's briefly go through this method to show you how it works and why the square root property was more efficient in this case.

Expanding the Equation

If we expand $(x-5)^2$, we get $(x-5)(x-5)$. Using the FOIL method:

• First: $x * x = x^2$
• Outer: $x * -5 = -5x$
• Inner: $-5 * x = -5x$
• Last: $-5 * -5 = 25$

So, $(x-5)^2 = x^2 - 10x + 25$. Our equation now becomes:

$x^2 - 10x + 25 = 1$

Rearranging and Simplifying

To solve this quadratic equation, we need to set it equal to zero. Subtract 1 from both sides:

$x^2 - 10x + 24 = 0$

Factoring the Quadratic

Now, we need to factor the quadratic expression. We're looking for two numbers that multiply to 24 and add up to -10. Those numbers are -6 and -4. So, we can factor the equation as:

$(x - 6)(x - 4) = 0$

Solving for x

Using the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero, we have:

• $x - 6 = 0$ or $x - 4 = 0$

Solving these gives us:

• $x = 6$ or $x = 4$

As you can see, we arrive at the same solutions, but the process is a bit longer and involves more steps. This is why recognizing the perfect square and using the square root property is often a more efficient approach for this type of problem.

Key Takeaways

Alright, guys, let's wrap things up with the key takeaways from this problem:

1. Square Root Property: Remember the square root property! If $(x-a)^2 = b$, then $x-a = \sqrt{b}$ or $x-a = -\sqrt{b}$. This is a powerful tool for solving equations with perfect squares.
2. Two Solutions: Quadratic equations often have two solutions. Don't forget to consider both the positive and negative square roots.
3. Check Your Answers: Always, always, always check your solutions by plugging them back into the original equation. This helps prevent errors.
4. Alternative Methods: While the square root property was efficient here, knowing how to expand and factor is also important for other types of quadratic equations.
5. Understanding Incorrect Options: Analyzing why other answer choices are wrong can deepen your understanding of the problem and the solution process.

Conclusion

So, there you have it! We've successfully solved the equation $(x-5)^2 = 1$ and found that the solutions are $x = 4$ and $x = 6$. We walked through the square root property method, checked our answers, discussed why other options were incorrect, and even touched on the alternative method of expanding and factoring. I hope this breakdown has been helpful and has boosted your confidence in solving similar problems. Keep practicing, and you'll become a math whiz in no time! Remember, math can be fun, especially when you break it down step by step. Keep up the great work, guys! And if you have any questions or want to explore more problems, just let me know. Happy solving!