Simplify (x+7)^2-(x-5)^2: Step-by-Step Guide

Hey guys! Today, we're diving into a classic algebraic problem: simplifying the expression $(x+7)^2 - (x-5)^2$. This might look a bit intimidating at first glance, but don't worry, we'll break it down step-by-step. We'll explore two main methods to tackle this: the direct expansion method and a slick shortcut using the difference of squares pattern. So, grab your pencils, and let's get started!

Understanding the Problem: What are we trying to do?

Before we jump into the solution, let's clarify what it means to "simplify completely." In algebra, simplifying means to rewrite an expression in its most basic form. This usually involves expanding any brackets, combining like terms, and ensuring there are no unnecessary operations left to perform. In our case, we have an expression involving squares of binomials (expressions with two terms) and a subtraction. Our goal is to expand these squares, simplify the resulting expression, and arrive at a concise answer.

At its core, this problem tests our understanding of algebraic expansion and simplification. Expanding expressions like $(x+7)^2$ involves multiplying the binomial by itself: $(x+7)(x+7)$. We use the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last) to ensure each term in the first binomial is multiplied by each term in the second. Simplifying then involves combining like terms – terms with the same variable raised to the same power (e.g., $3x$ and $5x$ are like terms, while $3x$ and $5x^2$ are not).

The expression $(x+7)^2-(x-5)^2$ also hints at a special algebraic pattern known as the difference of squares. Recognizing this pattern can provide a much faster route to the simplified answer. The difference of squares pattern states that $a^2 - b^2 = (a + b)(a - b)$. We'll explore how this applies to our problem later on. So, keep this in mind as we go through the direct expansion method first. Understanding these core concepts is key to unlocking not just this problem, but a whole range of algebraic challenges!

Method 1: Direct Expansion - A Detailed Walkthrough

The most straightforward way to simplify $(x+7)^2 - (x-5)^2$ is to expand each squared term individually and then combine like terms. Let's break this down into smaller, manageable steps:

Step 1: Expanding $(x+7)^2$

Remember that squaring a binomial means multiplying it by itself: $(x+7)^2 = (x+7)(x+7)$. Now, we use the distributive property (FOIL) to multiply the two binomials:

• First: $x * x = x^2$
• Outer: $x * 7 = 7x$
• Inner: $7 * x = 7x$
• Last: $7 * 7 = 49$

Combining these terms, we get: $x^2 + 7x + 7x + 49$. Simplifying further by combining the like terms (the $7x$ terms), we have: $x^2 + 14x + 49$.

Step 2: Expanding $(x-5)^2$

We follow the same process to expand $(x-5)^2 = (x-5)(x-5)$:

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

Combining these terms gives us: $x^2 - 5x - 5x + 25$. Simplifying by combining the like terms, we get: $x^2 - 10x + 25$.

Step 3: Subtracting the Expanded Expressions

Now we have the expanded forms of both squared terms: $(x+7)^2 = x^2 + 14x + 49$ and $(x-5)^2 = x^2 - 10x + 25$. The original problem asks us to subtract the second expression from the first: $(x^2 + 14x + 49) - (x^2 - 10x + 25)$.

This is a crucial step where we need to be careful with the signs. Subtracting an entire expression means subtracting each term within it. To avoid errors, it's helpful to distribute the negative sign:

$x^2 + 14x + 49 - x^2 + 10x - 25$.

Notice how the signs of the terms inside the second parenthesis have changed: $x^2$ became $-x^2$, $-10x$ became $+10x$, and $+25$ became $-25$.

Step 4: Combining Like Terms

Finally, we combine the like terms in the resulting expression: $x^2 + 14x + 49 - x^2 + 10x - 25$.

• $x^2$ terms: $x^2 - x^2 = 0$ (These cancel out!)
• $x$ terms: $14x + 10x = 24x$
• Constant terms: $49 - 25 = 24$

Therefore, after combining like terms, we are left with: $24x + 24$. This is the simplified form of the original expression. Direct expansion might seem lengthy, but it's a reliable method to solve such problems. Now, let's explore the quicker way using the difference of squares.

Method 2: The Difference of Squares Shortcut - A Smarter Approach

As we touched upon earlier, the expression $(x+7)^2 - (x-5)^2$ perfectly fits the difference of squares pattern: $a^2 - b^2 = (a + b)(a - b)$. Recognizing this pattern can save us a lot of time and effort. Let's see how it works in our case:

Step 1: Identifying 'a' and 'b'

In our expression, $(x+7)^2 - (x-5)^2$, we can identify:

• $a = (x+7)$
• $b = (x-5)$

We are essentially subtracting the square of 'b' from the square of 'a'.

Step 2: Applying the Difference of Squares Formula

Now, we can directly apply the formula: $a^2 - b^2 = (a + b)(a - b)$. Substituting our values for 'a' and 'b', we get:

$((x+7) + (x-5))((x+7) - (x-5))$

Notice how we've replaced $a^2 - b^2$ with a product of two binomial expressions. This is the key to simplifying using this method.

Step 3: Simplifying the Factors

Now we need to simplify the expressions within the parentheses. Let's start with the first factor:

$(x+7) + (x-5) = x + 7 + x - 5 = 2x + 2$

We simply combined the like terms ($x$ and $x$, and $7$ and $-5$).

Now, let's simplify the second factor:

$(x+7) - (x-5) = x + 7 - x + 5 = 12$

Here, we distributed the negative sign carefully: $-(x-5) = -x + 5$. Then, we combined like terms. Notice that the $x$ terms canceled out in this case.

Step 4: Multiplying the Simplified Factors

Now we have the simplified factors: $(2x + 2)$ and $(12)$. We need to multiply them together:

$(2x + 2)(12) = 12(2x + 2)$

We can distribute the $12$ across the terms inside the parenthesis:

$12 * 2x + 12 * 2 = 24x + 24$

And there we have it! We arrived at the same simplified expression, $24x + 24$, but with significantly fewer steps compared to the direct expansion method. The difference of squares pattern is a powerful tool for simplifying expressions like this.

Comparing the Methods: Which One is Best?

We've explored two different ways to simplify $(x+7)^2 - (x-5)^2$: direct expansion and the difference of squares shortcut. So, which method is better?

Direct Expansion:

• Pros: This method is very reliable and works for any expression involving squared binomials. It doesn't require recognizing any specific patterns.
• Cons: It can be more time-consuming and involves more steps, increasing the chances of making a small arithmetic error.

Difference of Squares:

• Pros: This method is significantly faster and more efficient if you recognize the pattern. It involves fewer steps and less computation.
• Cons: It only works for expressions that fit the difference of squares pattern. If you don't recognize the pattern, you can't use this method.

Which one to choose?

• If you're comfortable with algebraic manipulation and want a guaranteed method, direct expansion is a solid choice.
• If you can spot the difference of squares pattern, the shortcut is much faster and more elegant.

In general, recognizing patterns in mathematics is a valuable skill. It not only saves time but also deepens your understanding of the underlying concepts. In our case, recognizing the difference of squares allowed us to transform a complex-looking expression into a much simpler one.

Common Mistakes to Avoid

Simplifying algebraic expressions can be tricky, and there are a few common mistakes that students often make. Let's highlight some of these to help you avoid them:

1. Incorrectly Expanding Binomials: A frequent mistake is to simply square each term inside the parenthesis, for example, saying $(x+7)^2 = x^2 + 49$. Remember, squaring a binomial means multiplying it by itself, requiring the use of the distributive property (FOIL).
2. Sign Errors During Subtraction: When subtracting an entire expression, it's crucial to distribute the negative sign to every term inside the parenthesis. Forgetting to do this is a very common error. For example, in our problem, it's important to remember that $-(x^2 - 10x + 25) = -x^2 + 10x - 25$.
3. Combining Unlike Terms: Only like terms (terms with the same variable raised to the same power) can be combined. For instance, you can combine $3x$ and $5x$, but you cannot combine $3x$ and $5x^2$.
4. Forgetting the Order of Operations: Remember the order of operations (PEMDAS/BODMAS). Exponents (squaring in this case) should be performed before addition or subtraction.
5. Not Simplifying Completely: Make sure you've combined all possible like terms and that your final answer is in its simplest form. Leaving an expression like $24x + 24$ unsimplified would be considered incomplete.

By being aware of these common pitfalls, you can significantly improve your accuracy and confidence in simplifying algebraic expressions.

Practice Makes Perfect: Try These Problems!

To truly master the techniques we've discussed, it's essential to practice! Here are a few similar problems for you to try. Work through them using both the direct expansion method and the difference of squares shortcut (if applicable) to solidify your understanding:

1. $(y+3)^2 - (y-2)^2$
2. $(2a-1)^2 - (a+4)^2
3. $(m+5)^2 - (m-5)^2$

For each problem, make sure to:

• Expand the squared terms correctly using the distributive property.
• Pay close attention to signs, especially when subtracting expressions.
• Combine like terms carefully.
• Double-check your work to avoid common mistakes.

The more you practice, the more comfortable and proficient you'll become with simplifying algebraic expressions. Remember, math is like any other skill – it improves with consistent effort and dedication. So, don't be afraid to tackle challenging problems, and keep practicing! You've got this!

Conclusion: Mastering Algebraic Simplification

Alright, guys, we've covered a lot in this guide! We've explored how to simplify the expression $(x+7)^2 - (x-5)^2$ using two different methods: direct expansion and the difference of squares shortcut. We've also discussed the importance of recognizing algebraic patterns, common mistakes to avoid, and the value of practice. Simplifying algebraic expressions is a fundamental skill in mathematics, and mastering it will open doors to more advanced concepts. Whether you prefer the reliability of direct expansion or the efficiency of the difference of squares, the key is to understand the underlying principles and apply them carefully.

Remember, practice is crucial for building confidence and fluency in algebra. Work through the practice problems we provided, and don't hesitate to seek help or clarification when needed. Math can be challenging, but it's also incredibly rewarding when you grasp a new concept or solve a tricky problem. Keep exploring, keep learning, and keep simplifying! You're well on your way to becoming an algebra ace!