Let the number of rows be $r$. According to the problem, the number of students in each row is $3r - 1$.

Step 1: Form the equation

The total number of students is given by: $r \times (3r - 1) = 140$

Expanding this equation, we get: $3r^2 - r = 140$

Step 2: Rearrange the equation

Rewrite the equation as: $3r^2 - r - 140 = 0$

Step 3: Solve the quadratic equation

To solve $3r^2 - r - 140 = 0$, we use the quadratic formula: $r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 3$, $b = -1$, and $c = -140$.

Step 4: Calculate the discriminant

$b^2 - 4ac = (-1)^2 - 4(3)(-140) = 1 + 1680 = 1681$

Step 5: Find the roots

$r = \frac{-(-1) \pm \sqrt{1681}}{2 \times 3} = \frac{1 \pm 41}{6}$ Thus, the two solutions for $r$ are: $r = \frac{1 + 41}{6} = 7 \quad \text{and} \quad r = \frac{1 - 41}{6} = -\frac{40}{6}$

Since the number of rows cannot be negative, we choose $r = 7$.

Step 6: Calculate the number of students in each row

The number of students in each row is: $3r - 1 = 3(7) - 1 = 21 - 1 = 20$

Therefore, the number of students in each row is 20.

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. What would the number of students per row be if the total number of students increased to 180, following the same arrangement pattern?
2. How would the equation change if the number of students in a row was two less than twice the number of rows?
3. If there were 10 fewer students in the class, how would that affect the number of rows needed?
4. Can you derive a general formula for the number of students per row if the number of rows is $r$?
5. What other quadratic equations can arise from arranging students in different patterns?

Tip: When solving quadratic equations, always check if the discriminant is a perfect square to determine if the roots are rational numbers.