Verified Answer :

  • Expand the product \((9t - 4)(-9t - 4)\) using the distributive property (FOIL method) or algebraic manipulation.
  • Multiply the terms: \((9t \times -9t) + (9t \times -4) + (-4 \times -9t) + (-4 \times -4)\) results in \(-81t^2 - 36t + 36t + 16\).
  • Simplify the expression by combining the middle terms, which cancel each other out.
  • The final product is \(\boxed{-81t^2 + 16}\).

Explanation

  1. Analysis of the Expression

We want to expand the expression \((9t - 4)(-9t - 4)\). The expression is a product of two binomials involving a variable \(t\) and constant terms. We can expand this using two primary methods: applying the difference of squares formula after a small algebraic adjustment, or using the distributive property, commonly known as the FOIL method.

  1. Expansion via FOIL Method

To expand \((9t - 4)(-9t - 4)\) using the distributive property, we multiply each term in the first binomial by each term in the second binomial:

  • Multiply the first terms: \((9t) \times (-9t) = -81t^2\)
  • Multiply the outer terms: \((9t) \times (-4) = -36t\)
  • Multiply the inner terms: \((-4) \times (-9t) = 36t\)
  • Multiply the last terms: \((-4) \times (-4) = 16\)

Adding these results together:
\(-81t^2 - 36t + 36t + 16\)

Observe that the linear terms \(-36t\) and \(36t\) sum to zero:
\(-81t^2 + 0 + 16 = -81t^2 + 16\)

  1. Expansion via Difference of Squares Formula

Alternatively, we can rewrite the expression to use the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\).
First, notice that the second binomial \((-9t - 4)\) can be written as \(-(9t + 4)\).
So, \((9t - 4)(-9t - 4) = (9t - 4) \times -(9t + 4) = -(9t - 4)(9t + 4)\).
Applying the difference of squares formula with \(a = 9t\) and \(b = 4\):
\((9t - 4)(9t + 4) = (9t)^2 - (4)^2 = 81t^2 - 16\).
Distributing the negative sign back into the result:
\(-(81t^2 - 16) = -81t^2 + 16\).

  1. Conclusion

We have calculated that the product of \((9t - 4)(-9t - 4)\) is \(-81t^2 + 16\). Comparing this to the given options:
○ \(-81t^2 - 16\)
○ \(-81t^2 + 16\)
○ \(-81t^2 - 72t + 16\)
○ \(-81t^2 + 72t + 16\)

The correct option is \(-81t^2 + 16\).

Examples

When designing a physical structure, such as a bridge or a mechanical arm, engineers often need to calculate force distributions expressed as quadratic equations. Expanding such products is a fundamental skill in structural analysis. For example, if the stress on a beam is modeled by two intersecting components, multiplying these terms helps determine the overall resultant force, ensuring the structure can support the intended load effectively. Understanding how to expand binomial products allows engineers to simplify complex stress calculations into manageable, solvable forms.

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