Let's define the problem using variables:

• Let $r$ be the number of rows.
• Let $s$ be the number of students in each row.

We are given two pieces of information:

1. The total number of students is 494.
2. The number of students in a row ($s$) is one less than three times the number of rows ($r$). This can be written as: $s = 3r - 1$

Since the total number of students is the product of the number of rows and the number of students per row, we have: $r \times s = 494$

Substitute $s = 3r - 1$ into this equation: $r \times (3r - 1) = 494$

Expanding this gives: $3r^2 - r = 494$

Rearranging the terms, we get a quadratic equation: $3r^2 - r - 494 = 0$

Now, we can solve this quadratic equation using the quadratic formula: $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 3$, $b = -1$, and $c = -494$.

Substitute these values: $r = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 3 \cdot (-494)}}{2 \cdot 3}$ $r = \frac{1 \pm \sqrt{1 + 5928}}{6}$ $r = \frac{1 \pm \sqrt{5929}}{6}$ $r = \frac{1 \pm 77}{6}$

This results in two possible values for $r$: $r = \frac{1 + 77}{6} = 13 \quad \text{and} \quad r = \frac{1 - 77}{6} = -\frac{76}{6}$

Since the number of rows cannot be negative, we take $r = 13$.

Now, calculate $s$: $s = 3r - 1 = 3(13) - 1 = 39 - 1 = 38$

Final Answer:

The number of students in each row is 38.

Would you like a detailed explanation of the quadratic solving process or have any other questions?

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Here are 5 related questions for further practice:

1. If the number of students was increased to 550 with the same arrangement rule, how many rows would there be?
2. What is the sum of the roots of the quadratic equation $3r^2 - r - 494 = 0$?
3. If the number of rows was doubled while keeping the same rule, what would be the total number of students?
4. How many rows are needed if each row contains 4 more students than twice the number of rows?
5. What are the factors of the quadratic equation $3r^2 - r - 494 = 0$?

Tip: When solving quadratic equations, always check for factoring possibilities before using the quadratic formula—it can sometimes make the solution faster!