A class of 234 students are arranged in rows such that the number of students in a row is one less than thrice the number of rows. Find the number of students in each row.

Let the number of rows be denoted by \( r \), and the number of students in each row be \( 3r - 1 \) (since the number of students in a row is one less than three times the number of rows).

The total number of students is given as 234, so we can set up the following equation to model the situation:

\[
r \times (3r - 1) = 234
\]

Expanding the equation:

\[
3r^2 - r = 234
\]

Rearranging this into a standard quadratic form:

\[
3r^2 - r - 234 = 0
\]

Now, we can solve this quadratic equation using the quadratic formula, which is:

\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( 3r^2 - r - 234 = 0 \), the coefficients are:
- \( a = 3 \)
- \( b = -1 \)
- \( c = -234 \)

Substituting these into the quadratic formula:

\[
r = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-234)}}{2(3)}
\]

\[
r = \frac{1 \pm \sqrt{1 + 2808}}{6}
\]

\[
r = \frac{1 \pm \sqrt{2809}}{6}
\]

\[
r = \frac{1 \pm 53}{6}
\]

So, the two possible solutions are:

\[
r = \frac{1 + 53}{6} = \frac{54}{6} = 9
\]

or

\[
r = \frac{1 - 53}{6} = \frac{-52}{6} \approx -8.67
\]

Since the number of rows must be a positive integer, we discard the negative solution. Therefore, the number of rows is \( r = 9 \).

The number of students in each row is:

\[
3r - 1 = 3(9) - 1 = 27 - 1 = 26
\]

Thus, the number of students in each row is **26**.

Would you like any further details or explanations?

Here are 5 related questions:
1. How do you determine the total number of students in a different arrangement?
2. How do you solve other types of quadratic equations?
3. How would the solution change if the number of students was different?
4. What is the significance of the discriminant in solving quadratics?
5. How can you verify the solution by plugging back into the original equation?

**Tip:** When solving quadratics, always check both possible solutions, but choose the one that makes sense in the context of the problem.

A class of 234 students are arranged in rows such that the number of students in a row is one less than thrice the number of rows. Find the number of students in each row.

This problem involves arranging 234 students into rows where the number of students in each row is one less than three times the number of rows. Learn how to set up and solve a quadratic equation step-by-step. Suitable for Grades 8-10 students learning algebra and quadratic equations.

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