a. The average \( x \) of the three quizzes can be calculated using the formula:
\[ x = \frac{q1 + q2 + q3}{3} \]
where \( q1 \), \( q2 \), and \( q3 \) are the scores of the three quizzes.
b. William earned scores of 85 and 88 on the first two quizzes, so we can substitute these values into the average formula and set it equal to 90 to find the score he needs on the third quiz, \( q3 \):
\[ 90 = \frac{85 + 88 + q3}{3} \]
To solve for \( q3 \), first multiply both sides by 3:
\[ 270 = 85 + 88 + q3 \]
Next, combine the scores for the first two quizzes:
\[ 270 = 173 + q3 \]
Now, subtract 173 from both sides to isolate \( q3 \):
\[ q3 = 270 - 173 \]
Calculating this gives:
\[ q3 = 97 \]
Thus, the formula to determine the needed score on the third quiz is:
\[ q3 = 270 - (85 + 88) \]
And William needs to score 97 on the third quiz to achieve an average of 90.