To find the score the student needs on the fifth quiz to have an average of at least 84, let's follow these steps:

1. Let $x$ represent the score on the fifth quiz.

2. The formula for the average of five quiz scores is:

   $\text{Average} = \frac{\text{Sum of all quiz scores}}{\text{Number of quizzes}}$

   In this case, the number of quizzes is 5, and the average must be at least 84. So we set up the equation as:

   $\frac{86 + 81 + 95 + 75 + x}{5} \geq 84$

3. Add the known quiz scores:

   $86 + 81 + 95 + 75 = 337$

   So the equation becomes:

   $\frac{337 + x}{5} \geq 84$

4. Multiply both sides by 5 to eliminate the denominator:

   $337 + x \geq 420$

5. Solve for $x$ by subtracting 337 from both sides:

   $x \geq 420 - 337$

   $x \geq 83$

Therefore, the student must score at least 83 on the fifth quiz to have an average of at least 84.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How would the answer change if the desired average was 85 instead of 84?
2. What would the student’s average be if she scored exactly 83 on the fifth quiz?
3. How do weighted averages differ from simple averages?
4. How can we calculate the median quiz score with five scores?
5. What if there were more than five quizzes—how would we generalize this process?

Tip: Always double-check your final result by plugging it back into the original formula to verify its correctness.