Solving Equations: Find The Missing Expression

Nov 25, 2025 by Alex Johnson 47 views

Let's dive into the world of algebraic equations and tackle a fun problem: finding a missing expression. This is a common type of problem in mathematics, and mastering it can significantly boost your equation-solving skills. In this article, we'll break down a specific example step-by-step, making sure you understand the underlying concepts along the way. Our main goal is to determine what expression, when subtracted from $(-x^2 - 3x + 9)$ , results in $x^2 + 4x + 5$ . So, let's get started and unravel this algebraic puzzle together!

Understanding the Problem

Before we jump into the solution, let’s make sure we fully grasp what the problem is asking. We have an equation of the form:

$(-x^2 - 3x + 9) - (oxed{\phantom{blank}}) = x^2 + 4x + 5$

Our mission is to figure out what goes inside the box. In other words, we need to find the expression that, when subtracted from $(-x^2 - 3x + 9)$ , gives us $x^2 + 4x + 5$ . Think of it like a puzzle where we're missing a piece, and we need to find that piece to complete the picture. This involves using our knowledge of algebraic operations, particularly subtraction and rearranging equations. It's like being a detective, using clues to solve a mathematical mystery! To effectively solve equations, it’s crucial to understand the relationship between different parts and how they interact. By breaking down the problem into smaller, manageable steps, we can approach it with confidence and clarity.

Step-by-Step Solution

Now, let's break down how to solve this step-by-step. This approach isn't just about getting the right answer; it's about understanding the process so you can tackle similar problems with confidence.

1. Isolate the Missing Expression

The first thing we need to do is isolate the missing expression. Currently, it's being subtracted from the expression on the left side of the equation. To get it by itself, we can add it to both sides of the equation. This is a fundamental principle of algebra: what you do to one side, you must do to the other to maintain balance. So, adding the missing expression to both sides gives us:

$(-x^2 - 3x + 9) = x^2 + 4x + 5 + (oxed{\phantom{blank}})$

Notice how the missing expression is now on its own on the right side. However, we still need to get it completely isolated. To do this, we'll move the other terms on the right side to the left side.

2. Move Terms to the Other Side

To isolate the missing expression, we need to get rid of the $x^2 + 4x + 5$ on the right side. We can do this by subtracting $x^2 + 4x + 5$ from both sides of the equation. Remember, maintaining balance is key! This gives us:

$(-x^2 - 3x + 9) - (x^2 + 4x + 5) = (oxed{\phantom{blank}})$

Now we have the missing expression isolated on the right side, and we have an expression on the left side that we can simplify. This is a crucial step, as it sets us up to directly calculate the missing expression.

3. Simplify the Left Side

The next step is to simplify the left side of the equation by combining like terms. This involves distributing the negative sign and then adding or subtracting terms with the same variable and exponent. Let's break it down:

First, distribute the negative sign:

$-x^2 - 3x + 9 - x^2 - 4x - 5 = (oxed{\phantom{blank}})$

Now, combine like terms. We have $-x^2$ and $-x^2$ , which combine to $-2x^2$ . Then we have $-3x$ and $-4x$ , which combine to $-7x$ . Finally, we have $9$ and $-5$ , which combine to $4$ . So, the simplified equation is:

$-2x^2 - 7x + 4 = (oxed{\phantom{blank}})$

4. The Solution

We've done it! We've isolated the missing expression and simplified the equation. The expression that goes in the box is:

$-2x^2 - 7x + 4$

Therefore, the complete equation is:

$(-x^2 - 3x + 9) - (-2x^2 - 7x + 4) = x^2 + 4x + 5$

This is our final answer. We've successfully found the missing expression by carefully applying algebraic principles and simplifying the equation step-by-step. Understanding each step is more important than just getting the answer, as it allows you to apply these techniques to a wide range of problems.

Verification

It's always a good idea to check our answer to make sure it's correct. To do this, we can substitute the expression we found back into the original equation and see if it holds true. Let's substitute $-2x^2 - 7x + 4$ into the original equation:

$(-x^2 - 3x + 9) - (-2x^2 - 7x + 4) = x^2 + 4x + 5$

Now, let's simplify the left side:

$-x^2 - 3x + 9 + 2x^2 + 7x - 4 = x^2 + 4x + 5$

Combine like terms:

$(-x^2 + 2x^2) + (-3x + 7x) + (9 - 4) = x^2 + 4x + 5$

This simplifies to:

$x^2 + 4x + 5 = x^2 + 4x + 5$

The left side equals the right side, so our solution is correct! This verification step is crucial in ensuring the accuracy of our answer and reinforcing our understanding of the problem.

Key Concepts and Takeaways

Let's recap the key concepts and takeaways from this problem. This will help solidify your understanding and prepare you for future challenges.

Isolating the Unknown: The first crucial step in solving many algebraic equations is to isolate the variable or, in this case, the missing expression. We achieved this by performing the same operations on both sides of the equation, maintaining balance.
Combining Like Terms: Simplifying expressions by combining like terms is a fundamental skill. It involves adding or subtracting terms with the same variable and exponent. This makes the equation easier to work with and helps in arriving at the solution.
Distributive Property: We used the distributive property when we distributed the negative sign across the terms inside the parentheses. Understanding how to apply this property correctly is essential for simplifying algebraic expressions.
Verification: Always verify your solution by substituting it back into the original equation. This ensures accuracy and helps catch any mistakes made during the solving process.

By mastering these concepts, you'll be well-equipped to tackle a variety of algebraic problems. Remember, practice makes perfect, so keep solving equations and honing your skills!

Practice Problems

To further solidify your understanding, here are a couple of practice problems similar to the one we just solved:

Find the missing expression: $(2x^2 + 5x - 3) - (oxed{\phantom{blank}}) = x^2 - 2x + 1$
Determine the missing term: $(oxed{\phantom{blank}}) - (3x^2 - x + 4) = -x^2 + 2x - 2

Try solving these problems on your own, using the steps we outlined above. Remember to verify your solutions to ensure accuracy. Working through these practice problems will reinforce your understanding and build your confidence in solving algebraic equations.

Conclusion

In this article, we've walked through the process of finding a missing expression in an algebraic equation. We've seen how isolating the unknown, combining like terms, and verifying the solution are crucial steps in solving such problems. By understanding these concepts and practicing regularly, you can become proficient in solving a wide range of algebraic equations. Math is not just about finding answers; it's about understanding the process and logic behind the solutions. Keep practicing, keep exploring, and you'll continue to grow your mathematical skills.

For more resources on algebraic equations and practice problems, you can visit Khan Academy's Algebra Section.