Finding The Value Of G In The Equation (x+7)(x-4)=x^2+gx-28

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Introduction

In the realm of mathematics, equations serve as the bedrock for problem-solving and understanding the relationships between variables. Today, we're diving deep into a specific equation: (x+7)(x-4)=x^2+gx-28. Our primary goal is to uncover the value of 'g' that makes this equation hold true. This exploration will not only solidify our understanding of algebraic manipulation but also enhance our problem-solving skills. To truly grasp the solution, we'll embark on a journey through the intricacies of expanding expressions, simplifying equations, and ultimately isolating the variable 'g'. This journey will involve the application of fundamental algebraic principles, ensuring a comprehensive understanding of the underlying concepts. Let's delve into the fascinating world of equations and discover the value of 'g' that unlocks the truth behind this mathematical puzzle. Our approach will be methodical, ensuring that each step is clearly explained and justified, leading us to a confident and accurate solution. The beauty of mathematics lies in its ability to reveal hidden truths, and this equation is no exception. So, let's roll up our sleeves and embark on this mathematical adventure, where we'll unravel the mystery surrounding the value of 'g'. By the end of this exploration, you'll not only know the answer but also possess a deeper appreciation for the power and elegance of algebraic equations.

Expanding the Left Side of the Equation

The first step in our quest to find the value of 'g' is to expand the left side of the equation, which is (x+7)(x-4). This involves using the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). The FOIL method helps us systematically multiply each term in the first binomial by each term in the second binomial. Let's break it down:

  • First: Multiply the first terms of each binomial: x * x = x^2
  • Outer: Multiply the outer terms of the binomials: x * -4 = -4x
  • Inner: Multiply the inner terms of the binomials: 7 * x = 7x
  • Last: Multiply the last terms of each binomial: 7 * -4 = -28

Now, let's combine these results:

x^2 - 4x + 7x - 28

Next, we simplify this expression by combining like terms, specifically the '-4x' and '7x' terms. This gives us:

x^2 + 3x - 28

This expanded form of the left side of the equation is crucial for our next step, where we'll compare it to the right side and isolate the variable 'g'. This meticulous expansion process demonstrates the importance of algebraic manipulation in solving equations. By carefully applying the distributive property and combining like terms, we've transformed the left side into a more manageable form, bringing us closer to our goal of finding the value of 'g'. The expanded form, x^2 + 3x - 28, now stands in stark contrast to the right side of the equation, setting the stage for a direct comparison that will reveal the solution we seek. This step-by-step approach ensures clarity and accuracy, allowing us to proceed with confidence in our mathematical journey.

Comparing Both Sides of the Equation

Now that we've expanded the left side of the equation, we have:

x^2 + 3x - 28

Our original equation was:

(x+7)(x-4) = x^2 + gx - 28

And we've transformed the left side to:

x^2 + 3x - 28 = x^2 + gx - 28

Now, we can directly compare both sides of the equation. Notice that the x^2 terms and the constant terms (-28) are the same on both sides. This simplifies our task significantly, as we can now focus solely on the 'x' terms. The left side has '3x', while the right side has 'gx'. For the equation to be true for all values of 'x', the coefficients of the 'x' terms must be equal. This is a fundamental principle of algebraic equations: if two polynomials are equal, their corresponding coefficients must be equal. In this case, the coefficient of 'x' on the left side is 3, and the coefficient of 'x' on the right side is 'g'. Therefore, we can confidently equate these coefficients to find the value of 'g'. This direct comparison highlights the power of algebraic manipulation in isolating variables and revealing their values. By carefully expanding and simplifying the equation, we've created a clear and straightforward path to the solution. The equality of the x^2 and constant terms acts as a validation of our previous steps, reinforcing the accuracy of our calculations. This methodical approach not only leads us to the correct answer but also deepens our understanding of the underlying mathematical principles.

Isolating and Determining the Value of g

From our comparison in the previous section, we have:

3x = gx

To isolate 'g', we need to divide both sides of the equation by 'x'. However, we must be mindful of a crucial caveat: we can only divide by 'x' if 'x' is not equal to zero. If 'x' were zero, we would be dividing by zero, which is undefined in mathematics. Assuming 'x' is not zero, we can proceed with the division:

(3x) / x = (gx) / x

This simplifies to:

3 = g

Therefore, the value of 'g' that makes the equation true is 3. This result is a direct consequence of the equality of coefficients, a powerful tool in solving algebraic equations. The process of isolating 'g' by dividing both sides by 'x' demonstrates the importance of understanding the rules of algebraic manipulation. The caveat regarding 'x' not being zero highlights the need for careful consideration of potential exceptions and limitations in mathematical operations. Our solution, g = 3, is not just a numerical answer; it's a testament to the logical and systematic approach we've employed throughout this exploration. This value of 'g' ensures that the equation (x+7)(x-4) = x^2 + gx - 28 holds true for all values of 'x' (except possibly x = 0, which we've addressed). This thorough analysis solidifies our understanding of the equation and the role of 'g' in maintaining its balance.

Conclusion

In conclusion, by carefully expanding the left side of the equation (x+7)(x-4), comparing it to the right side, and isolating the variable 'g', we have successfully determined that the value of g = 3 makes the equation true. This journey through algebraic manipulation has reinforced the importance of techniques like the FOIL method, combining like terms, and the principle of equality of coefficients. We've also highlighted the crucial consideration of potential exceptions, such as dividing by zero, in mathematical operations. This exploration serves as a valuable exercise in problem-solving, demonstrating how a methodical and systematic approach can unlock the solution to seemingly complex equations. The value of 'g' is not just a number; it's the key that unlocks the truth behind the equation, ensuring its validity and balance. Our journey has not only provided us with the answer but has also deepened our understanding of the underlying mathematical principles at play. The ability to manipulate equations, compare coefficients, and isolate variables is a fundamental skill in mathematics, and this exercise has honed our proficiency in these areas. So, the next time you encounter a similar equation, remember the steps we've taken here, and you'll be well-equipped to unravel its mysteries and discover the hidden values within.