Find The Value Of G In (x+7)(x-4) = X^2 + Gx - 28
In the realm of algebra, solving equations is a fundamental skill. Today, we embark on a journey to determine the value of 'g' that makes the equation (x+7)(x-4) = x^2 + gx - 28 true. This equation presents an intriguing challenge, requiring us to expand, simplify, and compare coefficients to unlock the mystery of 'g'.
Expanding and Simplifying the Equation
Our initial step involves expanding the left side of the equation, (x+7)(x-4). This can be achieved using the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last). Applying this method, we get:
(x+7)(x-4) = x(x) + x(-4) + 7(x) + 7(-4)
Simplifying this expression, we have:
x^2 - 4x + 7x - 28
Combining the like terms, specifically the 'x' terms, we arrive at:
x^2 + 3x - 28
Now, our equation takes the form:
x^2 + 3x - 28 = x^2 + gx - 28
Comparing Coefficients: The Key to Finding g
The next crucial step in our quest to find 'g' is to compare the coefficients of the corresponding terms on both sides of the equation. By aligning the terms with the same powers of 'x', we can establish a direct relationship between the coefficients.
On the left side of the equation, the coefficient of the 'x' term is 3. On the right side, the coefficient of the 'x' term is 'g'. For the equation to hold true for all values of 'x', the coefficients of the corresponding terms must be equal. This principle forms the bedrock of our solution.
Therefore, by equating the coefficients of the 'x' terms, we arrive at the following conclusion:
g = 3
This elegant solution reveals that the value of 'g' that makes the equation true is 3. By substituting g = 3 back into the original equation, we can verify our solution:
x^2 + 3x - 28 = x^2 + 3x - 28
The equation holds true, confirming the accuracy of our result. This exercise demonstrates the power of algebraic manipulation and the importance of comparing coefficients in solving equations.
Alternative Approaches and Deeper Insights
While we have successfully determined the value of 'g' by expanding and comparing coefficients, it's worth exploring alternative approaches that can provide deeper insights into the problem. One such approach involves recognizing the relationship between the roots of a quadratic equation and its coefficients.
Consider a general quadratic equation of the form:
ax^2 + bx + c = 0
The sum of the roots of this equation is given by -b/a, and the product of the roots is given by c/a. In our case, the equation (x+7)(x-4) = x^2 + gx - 28 can be rewritten as:
x^2 + 3x - 28 = 0
Comparing this equation with the general form, we have a = 1, b = 3, and c = -28. The roots of this equation are -7 and 4, which correspond to the values that make the factors (x+7) and (x-4) equal to zero.
The sum of the roots is -7 + 4 = -3, and the product of the roots is -7 * 4 = -28. These values align with the coefficients in the equation, further validating our understanding of the relationship between roots and coefficients.
Moreover, we can connect this problem to the concept of polynomial identities. A polynomial identity is an equation that holds true for all values of the variable. Our original equation, (x+7)(x-4) = x^2 + gx - 28, can be viewed as a polynomial identity if and only if g = 3. This perspective highlights the broader context of our solution within the realm of polynomial algebra.
The Significance of Understanding Quadratic Equations
The problem we have tackled underscores the significance of understanding quadratic equations. Quadratic equations are fundamental to various fields, including mathematics, physics, engineering, and economics. They arise in scenarios involving projectile motion, optimization problems, and curve fitting, among others.
A deep understanding of quadratic equations empowers us to model and solve real-world problems. The techniques we have employed, such as expanding, simplifying, comparing coefficients, and relating roots to coefficients, are valuable tools in the problem-solving arsenal.
Furthermore, the ability to manipulate and analyze quadratic equations lays the foundation for exploring higher-degree polynomials and more complex mathematical concepts. As we delve deeper into the realm of mathematics, the principles we have learned here will continue to serve us well.
Conclusion: A Triumphant Journey of Algebraic Exploration
In conclusion, we have successfully navigated the equation (x+7)(x-4) = x^2 + gx - 28 and determined that the value of 'g' that makes the equation true is 3. Our journey involved expanding, simplifying, comparing coefficients, and exploring alternative approaches that enriched our understanding of the problem.
This exercise has not only provided us with a specific solution but has also reinforced fundamental algebraic principles and highlighted the significance of quadratic equations in mathematics and beyond. As we continue our exploration of the mathematical landscape, the lessons learned here will undoubtedly illuminate our path.
This article explains how to find the value of g that makes the equation (x+7)(x-4) = x^2 + gx - 28 true. We'll explore the steps involved in solving this algebraic problem, focusing on expanding, simplifying, and comparing coefficients. Understanding these concepts is crucial for mastering algebra and solving similar equations.
Step-by-Step Solution to Determine the Value of g
To find the value of g, we need to manipulate the equation (x+7)(x-4) = x^2 + gx - 28 and isolate g. Here’s a breakdown of the steps:
1. Expand the Left Side of the Equation
First, we need to expand the left side of the equation, which is (x+7)(x-4). We can use the FOIL method (First, Outer, Inner, Last) to do this. The FOIL method ensures that we multiply each term in the first set of parentheses by each term in the second set:
- First: x * x = x^2
- Outer: x * -4 = -4x
- Inner: 7 * x = 7x
- Last: 7 * -4 = -28
So, expanding (x+7)(x-4) gives us:
x^2 - 4x + 7x - 28
2. Simplify the Expanded Expression
Next, we simplify the expanded expression by combining like terms. In this case, we can combine the -4x and 7x terms:
x^2 - 4x + 7x - 28 = x^2 + 3x - 28
Now our equation looks like this:
x^2 + 3x - 28 = x^2 + gx - 28
3. Compare Coefficients
Now we compare the coefficients of the terms on both sides of the equation. Specifically, we are interested in the coefficients of the x terms. On the left side, the coefficient of x is 3, and on the right side, the coefficient of x is g.
For the equation to be true, the coefficients of the corresponding terms must be equal. Therefore, we can set the coefficients of the x terms equal to each other:
3 = g
4. State the Solution
From the comparison of coefficients, we find that:
g = 3
This is the value of g that makes the equation (x+7)(x-4) = x^2 + gx - 28 true.
Why is Comparing Coefficients Important?
Comparing coefficients is a powerful technique in algebra. It allows us to solve for unknown variables in equations by ensuring that the terms on both sides of the equation match up. This method is particularly useful when dealing with polynomials, as it provides a straightforward way to equate expressions and find the values that make them equivalent.
Applications of Coefficient Comparison
- Solving Polynomial Equations: As demonstrated in this problem, comparing coefficients can help find unknown constants in polynomial equations.
- Partial Fraction Decomposition: In calculus and advanced algebra, this technique is used to break down rational expressions into simpler fractions.
- Verifying Identities: By comparing coefficients, we can verify if a given equation is an identity, meaning it holds true for all values of the variable.
Common Mistakes to Avoid
When solving equations like this, there are a few common mistakes to watch out for:
- Incorrect Expansion: Make sure to correctly apply the distributive property (FOIL method) when expanding expressions.
- Missing Terms: Ensure that you account for all terms when expanding and simplifying.
- Sign Errors: Pay close attention to signs, especially when dealing with negative numbers.
- Incorrect Comparison: Make sure you are comparing the coefficients of the corresponding terms.
Practice Problems to Enhance Understanding
To solidify your understanding of finding the value of g, try solving similar problems. Here are a few examples:
- Find k if (x-2)(x+5) = x^2 + kx - 10.
- Determine m if (x+3)(x-1) = x^2 + mx - 3.
- Solve for n in the equation (x-4)(x-4) = x^2 + nx + 16.
Working through these practice problems will help you become more comfortable with the process of expanding, simplifying, and comparing coefficients.
Real-World Applications of Algebra
Algebraic equations are not just abstract mathematical concepts; they have numerous real-world applications. Understanding algebra can help you solve problems in various fields, including:
- Engineering: Designing structures, circuits, and systems often involves algebraic equations.
- Physics: Modeling physical phenomena, such as motion and forces, relies on algebraic principles.
- Economics: Analyzing markets, predicting trends, and making financial decisions often require algebraic techniques.
- Computer Science: Developing algorithms, writing code, and analyzing data structures use algebraic concepts.
Conclusion: Mastering Algebra for Problem Solving
In conclusion, we have successfully found the value of g in the equation (x+7)(x-4) = x^2 + gx - 28 by expanding, simplifying, and comparing coefficients. This exercise highlights the importance of algebraic manipulation and the power of comparing coefficients in solving equations. By practicing these techniques and understanding their applications, you can enhance your problem-solving skills and tackle more complex algebraic challenges. Remember to avoid common mistakes and work through practice problems to solidify your understanding. Algebra is a fundamental tool that opens doors to various fields, and mastering it will serve you well in your academic and professional pursuits.
In this detailed guide, we will explore how to find the specific value of 'g' that makes the algebraic equation (x+7)(x-4) = x^2 + gx - 28 true. This equation is a prime example of how algebraic principles can be applied to solve for unknown variables by manipulating expressions and comparing coefficients. We'll break down the process into clear, manageable steps, ensuring a thorough understanding of the underlying concepts.
Unpacking the Equation: A Step-by-Step Approach
To successfully determine the value of 'g', we need to methodically approach the equation. This involves expanding the expressions, simplifying the result, and then comparing coefficients to isolate and solve for 'g'. Let's delve into each step.
Step 1: Expanding the Left-Hand Side
The initial step in solving this equation is to expand the left-hand side, which is the product of two binomials: (x+7)(x-4). To do this, we employ the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first binomial is multiplied by each term in the second binomial.
- First: Multiply the first terms in each binomial: x * x = x^2
- Outer: Multiply the outer terms in the binomials: x * -4 = -4x
- Inner: Multiply the inner terms in the binomials: 7 * x = 7x
- Last: Multiply the last terms in each binomial: 7 * -4 = -28
Combining these results gives us the expanded expression:
x^2 - 4x + 7x - 28
Step 2: Simplifying the Expanded Expression
Now that we have expanded the left-hand side, we need to simplify it by combining like terms. In this case, the like terms are the '-4x' and '7x' terms. Combining these gives us:
-4x + 7x = 3x
So, the simplified expression becomes:
x^2 + 3x - 28
Step 3: Equating the Simplified Expression to the Right-Hand Side
After simplifying the left-hand side, we can now rewrite the original equation as:
x^2 + 3x - 28 = x^2 + gx - 28
This step is crucial because it allows us to directly compare the coefficients of corresponding terms on both sides of the equation.
Step 4: Comparing Coefficients to Solve for 'g'
This is the pivotal step where we determine the value of 'g'. To do this, we compare the coefficients of the 'x' terms on both sides of the equation. The coefficient of 'x' on the left-hand side is '3', and the coefficient of 'x' on the right-hand side is 'g'.
For the equation to hold true for all values of 'x', the coefficients of the corresponding terms must be equal. Therefore, we can equate the coefficients:
3 = g
This immediately gives us the value of 'g'.
Step 5: Stating the Solution
From the comparison of coefficients, we have found that:
g = 3
This is the value of 'g' that makes the original equation (x+7)(x-4) = x^2 + gx - 28 true. This solution ensures that both sides of the equation are equivalent for all values of 'x'.
The Importance of Understanding Algebraic Manipulation
The process we've used to solve for 'g' highlights the importance of algebraic manipulation skills. These skills are fundamental to solving a wide range of mathematical problems, including those found in algebra, calculus, and beyond.
Key Algebraic Principles Used
- Distributive Property: This property allows us to expand expressions involving parentheses, as we did in Step 1.
- Combining Like Terms: This simplifies expressions by grouping terms with the same variable and exponent, as seen in Step 2.
- Equating Coefficients: This powerful technique allows us to solve for unknown variables by comparing corresponding terms in an equation, as demonstrated in Step 4.
Avoiding Common Mistakes in Algebraic Problem Solving
When working with algebraic equations, it's essential to be mindful of common mistakes that can lead to incorrect solutions. Here are a few to watch out for:
- Sign Errors: Pay close attention to signs, especially when distributing negative numbers or combining like terms.
- Incorrect Expansion: Ensure that you correctly apply the distributive property and multiply each term appropriately.
- Missing Terms: Be careful not to overlook any terms when expanding or simplifying expressions.
- Miscomparing Coefficients: Make sure you are comparing the coefficients of corresponding terms on both sides of the equation.
Practicing to Improve Algebraic Skills
The best way to improve your algebraic skills is through practice. Solving a variety of problems will help you become more comfortable with the techniques and concepts involved. Here are a few practice problems similar to the one we've solved:
- Find the value of 'k' in the equation (x-3)(x+2) = x^2 + kx - 6.
- Determine the value of 'm' in the equation (x+1)(x-5) = x^2 + mx - 5.
- Solve for 'n' in the equation (x-2)(x-2) = x^2 + nx + 4.
Real-World Applications of Algebraic Equations
Algebraic equations are not just theoretical exercises; they have numerous real-world applications across various fields. Understanding algebra is crucial for solving problems in:
- Physics: Modeling motion, forces, and energy often involves algebraic equations.
- Engineering: Designing structures, circuits, and systems requires algebraic calculations.
- Economics: Analyzing markets, making financial projections, and understanding economic models relies on algebra.
- Computer Science: Developing algorithms, writing code, and analyzing data structures use algebraic concepts.
Conclusion: Mastering Algebra for Success
In this comprehensive guide, we have thoroughly explored how to find the value of 'g' in the equation (x+7)(x-4) = x^2 + gx - 28. By systematically expanding, simplifying, and comparing coefficients, we determined that g = 3. This process underscores the significance of algebraic manipulation skills and the importance of avoiding common mistakes. With practice and a solid understanding of algebraic principles, you can confidently tackle a wide range of mathematical challenges. Algebra is a foundational tool that empowers you to succeed in various academic and professional pursuits.
