What value of $g$ makes the equation true?
$(x+7)(x-4)=x^2+g x-28$
We are given the equation \((x+7)(x-4) = x^2 + gx - 28\). The objective is to find the value of the constant \(g\) that makes this equation true for all values of \(x\). To do this, we need to expand the product on the left-hand side of the equation.
To expand \((x+7)(x-4)\), we use the distributive property (often called the FOIL method for binomials).
Multiplying the terms, we get:
\[ (x+7)(x-4) = x \cdot x + x \cdot (-4) + 7 \cdot x + 7 \cdot (-4) \]
\[ = x^2 - 4x + 7x - 28 \]
Combine the like terms, \(-4x\) and \(7x\), which result in \(3x\):
\[ = x^2 + 3x - 28 \]
Now, compare the expanded left-hand side, \(x^2 + 3x - 28\), with the right-hand side, \(x^2 + gx - 28\). For these two expressions to be identical for all values of \(x\), their corresponding coefficients must be equal. Comparing the linear terms, we have:
\[ 3x = gx \]
Dividing by \(x\) (assuming \(x
eq 0\)), we find:
\[ g = 3 \]
The constant terms, \(-28\) and \(-28\), are identical, which confirms our expansion is correct and consistent.
The value of \(g\) that satisfies the equation is \(3\). Comparing this result with the given options, this matches option (a). Thus, the value is \(\boxed{3}\).
Imagine you are building a rectangular garden with dimensions \(x+7\) feet and \(x-4\) feet. The area of the garden is given by the product of these dimensions, \((x+7)(x-4)\), which calculates the total square footage. Expanding this expression into the form \(x^2 + gx - 28\) helps in understanding how individual parts of the dimension lengths contribute to the total area, allowing landscapers or architects to efficiently calculate materials or space requirements for various garden designs.