Unlock Algebraic Mysteries: Solving For 'g'

When faced with an algebraic equation, it's natural to wonder about the specific values of variables that will make the entire statement true. In this case, we're presented with the equation $(x+7)(x-4)=x^2+9x-28$. While the equation is already presented in a seemingly solved form, the core question is to determine if there's a specific value for a variable, let's assume it's represented by '$g$' in a broader context or perhaps it's a typo and should be related to the coefficients or constants within the given equation. Often, these types of problems are designed to test your understanding of algebraic manipulation and the fundamental properties of equations. Let's break down the equation and explore how to approach this.

Understanding the Equation

The equation $(x+7)(x-4)=x^2+9x-28$ involves two sides. The left side is a product of two binomials, $(x+7)$ and $(x-4)$. The right side is a quadratic trinomial, $x^2+9x-28$. To determine if the equation holds true for all values of $x$, or if there's a specific condition to be met, we need to expand the left side and compare it to the right side.

We can expand the left side using the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last):

• First: $x * x = x^2$
• Outer: $x * (-4) = -4x$
• Inner: $7 * x = 7x$
• Last: $7 * (-4) = -28$

Combining these terms, we get: $x^2 - 4x + 7x - 28$.

Now, we simplify this expression by combining the like terms (the $x$ terms):

$x^2 + (-4x + 7x) - 28 = x^2 + 3x - 28$.

So, the expanded form of the left side of the equation is $x^2 + 3x - 28$.

Now, let's compare this to the right side of the original equation, which is $x^2 + 9x - 28$.

Our expanded left side is $x^2 + 3x - 28$, and the given right side is $x^2 + 9x - 28$.

For the equation $(x+7)(x-4)=x^2+9x-28$ to be true, the expanded left side must equal the right side: $x^2 + 3x - 28 = x^2 + 9x - 28$.

Identifying the Discrepancy

Let's analyze the equation $x^2 + 3x - 28 = x^2 + 9x - 28$.

We can see that the $x^2$ terms are identical on both sides, and the constant terms (-28) are also identical. The only difference lies in the coefficient of the $x$ term. On the left side, it's $+3x$, and on the right side, it's $+9x$.

To solve for $x$ (if that were the goal, but our problem is about finding '$g$'), we would subtract $x^2$ from both sides:

$3x - 28 = 9x - 28$.

Then, we would add 28 to both sides:

$3x = 9x$.

Finally, we would subtract $3x$ from both sides:

$0 = 6x$.

Dividing by 6, we get $x=0$.

This means that the original equation $(x+7)(x-4)=x^2+9x-28$ is only true when $x=0$. However, the question asks for the value of '$g$' that makes the equation true. This suggests that '$g$' is not a variable like $x$ that we solve for within the structure of the equation as presented. Instead, '$g$' likely represents a missing constant or coefficient that, if correctly identified, would make the equation an identity (true for all $x$) or would represent a specific condition.

Reinterpreting the Question: What is '$g$'?

Given the standard format of algebraic problems, it's highly probable that the question intends to imply that the equation should be true, and the value of '$g$' is what's missing or incorrect. If the problem intended for the equation to be an identity (true for all values of $x$), then the coefficients on both sides must match.

Let's assume the equation was meant to be $(x+7)(x-4)=x^2+gx-28$. In this hypothetical scenario, we would expand the left side: $x^2 + 3x - 28$. For this to equal $x^2+gx-28$, the coefficient of the $x$ term must match. Therefore, $g$ would have to be 3.

Alternatively, if the equation was presented as $(x+a)(x+b)=x^2+9x-28$, we would be looking for factors of -28 that add up to 9. These numbers are 11.61 and -2.61, not integer values. If the factors were $(x+7)$ and $(x+g)$, then $(x+7)(x+g) = x^2 + gx + 7x + 7g = x^2 + (g+7)x + 7g$. For this to equal $x^2+9x-28$, we would need:

1. $g+7 = 9 ightarrow g = 2$
2. $7g = -28 ightarrow g = -4$

These two conditions for $g$ contradict each other ($g=2$ and $g=-4$), indicating that the original premise of $(x+7)(x+g)=x^2+9x-28$ is not consistent with the structure of the problem if we assume the problem implies a correct and consistent algebraic identity.

Let's return to the exact wording: "What value of $g$ makes the equation true? $(x+7)(x-4)=x^2+9 x-28$"

The most straightforward interpretation is that the '$g$' is part of the original equation, but it was written as a '9' in the question. It's a common scenario in math problems that a variable is intended but a number is written instead, or vice-versa. If we assume that the '$9$' in $x^2+9x-28$ was meant to be '$g$', then the equation would be $(x+7)(x-4)=x^2+gx-28$.

As we calculated earlier, expanding $(x+7)(x-4)$ yields $x^2 + 3x - 28$.

For this to be equal to $x^2+gx-28$, we must equate the coefficients of the $x$ term:

$3x = gx$

This implies $g=3$.

Conclusion: The Value of '$g$'

Based on the standard conventions of algebraic problem-solving and the given structure, the most logical conclusion is that the digit '9' in the equation $(x+7)(x-4)=x^2+9x-28$ was intended to be the variable '$g$'. Thus, the equation was meant to be written as $(x+7)(x-4)=x^2+gx-28$.

When we expand the left side, $(x+7)(x-4)$, we perform the multiplication:

$x imes x = x^2$ $x imes -4 = -4x$ $7 imes x = 7x$ $7 imes -4 = -28$

Combining these terms gives us $x^2 - 4x + 7x - 28$, which simplifies to $x^2 + 3x - 28$.

Now, we set this equal to the right side of the intended equation: $x^2 + 3x - 28 = x^2 + gx - 28$.

To make this equation true for all values of $x$, the coefficients of corresponding terms must be equal. The $x^2$ terms match, and the constant terms (-28) match. Therefore, the coefficients of the $x$ terms must also match:

$3x = gx$

This implies that $g=3$.

So, the value of $g$ that makes the equation $(x+7)(x-4)=x^2+gx-28$ true is $3$.

If the question is strictly interpreted as