Verified Answer :

  • Expand the expression \((7x - 4)(5x + 3)\) using the distributive property.
  • Combine the resulting linear terms \(21x\) and \(-20x\) into \(x\).
  • Simplify the expression to \(35x^2 + x - 12\).
  • Match the resulting constant term with the form \(Fx^2 + Gx + H\) to find \(\boxed{-12}\).

Explanation

  1. Analysis of the Algebraic Expression

To find the value of \(H\) in the equation \((7x - 4)(5x + 3) = Fx^2 + Gx + H\), we must first expand the expression on the left side. This involves multiplying two binomials, \((7x - 4)\) and \((5x + 3)\), which represent the product of two linear terms. Expanding this will yield a quadratic expression of the form \(Ax^2 + Bx + C\), where we can directly identify the coefficients \(F\), \(G\), and \(H\).

  1. Expanding and Simplifying the Expression

We apply the distributive property (often referred to as the FOIL method) to expand the product \((7x - 4)(5x + 3)\).

The FOIL method consists of:

  • First terms: \((7x) \times (5x) = 35x^2\)
  • Outer terms: \((7x) \times (3) = 21x\)
  • Inner terms: \((-4) \times (5x) = -20x\)
  • Last terms: \((-4) \times (3) = -12\)

Combining these results, we get the expression:
\[35x^2 + 21x - 20x - 12\]
By combining the like linear terms (\(21x\) and \(-20x\)), we simplify the expression to:
\[35x^2 + x - 12\]

  1. Identifying the Constant \(H\)

Now that we have expanded the left side as \(35x^2 + x - 12\), we compare it to the right side of the given equation: \(Fx^2 + Gx + H\).
By equating corresponding coefficients:

  • \(F = 35\)
  • \(G = 1\)
  • \(H = -12\)

Thus, the value of the constant term \(H\) is \(-12\).

Examples

Understanding how to expand binomials is crucial for simplifying complex expressions in various fields, such as physics and economics. For example, if you are modeling the area of a rectangle whose sides increase over time, you might express the sides as \(L(t) = 7t - 4\) and \(W(t) = 5t + 3\). The area, \(A(t) = L(t) \times W(t)\), can be expanded to understand the growth pattern of the area over time, where the constant term often represents the base value or an offset in the model. Mastering this helps in predicting how different variables interact to form quadratic relationships.

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