Verified Answer :

  • A rational expression \(\frac{P(x)}{Q(x)}\) equals zero if \(P(x) = 0\) and \(Q(x) eq 0\).
  • Evaluating \(\frac{7x-5}{x^2+10}\) at \(x=2\) gives \(\frac{9}{14} eq 0\).
  • Evaluating \(\frac{-3x+6}{8x-9}\) at \(x=2\) gives \(\frac{0}{7} = 0\).
  • Evaluating \(\frac{-5x+8}{x-2}\) at \(x=2\) results in division by zero, which is undefined.
  • The correct expression is \(\boxed{\frac{-3x+6}{8x-9}}\).

Explanation

  1. Analysis of the Problem

To determine which rational expression equals zero at \(x=2\), we examine each expression \(\frac{P(x)}{Q(x)}\). A rational expression is equal to zero if and only if its numerator \(P(x) = 0\) and its denominator \(Q(x) eq 0\). Evaluating the expressions at \(x=2\) involves substituting \(2\) for \(x\) and checking these two conditions.

  1. Evaluating the First Expression

We test the first expression, \(\frac{7x-5}{x^2+10}\):

  • Numerator: \(7(2) - 5 = 14 - 5 = 9\).
  • Since the numerator is not equal to zero, this expression does not equal zero at \(x=2\).
  1. Evaluating the Second Expression

We test the second expression, \(\frac{-3x+6}{8x-9}\):

  • Numerator: \(-3(2) + 6 = -6 + 6 = 0\).
  • Denominator: \(8(2) - 9 = 16 - 9 = 7\).
  • Since the numerator is zero and the denominator is non-zero, this expression equals zero at \(x=2\).
  1. Evaluating the Third Expression

We test the third expression, \(\frac{-5x+8}{x-2}\):

  • Numerator: \(-5(2) + 8 = -10 + 8 = -2\).
  • Denominator: \(2 - 2 = 0\).
  • Because the denominator is zero, the expression is undefined at \(x=2\). It does not equal zero.
  1. Conclusion

Based on our evaluations, the rational expression \(\frac{-3x+6}{8x-9}\) is the only one that satisfies the condition of having a value of zero at \(x=2\). Thus, the correct choice is \(\frac{-3x+6}{8x-9}\). Therefore, the final answer is \(\boxed{\frac{-3x+6}{8x-9}}\).

Examples

In engineering and physics, rational functions often describe the relationship between two physical quantities, such as velocity or acceleration. For instance, if you are designing a braking system, a rational expression might model the force applied to a rotor relative to the speed of the wheel. Identifying the "zeros" of these functions allows engineers to find critical points—such as the exact speed at which the net force on a system becomes zero—which is vital for ensuring stability and optimal performance in mechanical designs.

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