A class of 352 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 \( x \). The number of students in each row is then \( 3x - 1 \), according to the problem.

The total number of students is given by the product of the number of rows and the number of students per row. Therefore, we can set up the following equation:

\[
x \times (3x - 1) = 352
\]

Expanding the equation:

\[
3x^2 - x = 352
\]

Rearranging the equation to standard quadratic form:

\[
3x^2 - x - 352 = 0
\]

Now, we'll solve this quadratic equation using the quadratic formula. The quadratic formula is:

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

For the equation \( 3x^2 - x - 352 = 0 \), the coefficients are:
- \( a = 3 \)
- \( b = -1 \)
- \( c = -352 \)

Substituting into the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-352)}}{2(3)}
\]

\[
x = \frac{1 \pm \sqrt{1 + 4224}}{6}
\]

\[
x = \frac{1 \pm \sqrt{4225}}{6}
\]

\[
x = \frac{1 \pm 65}{6}
\]

This gives two possible solutions:

\[
x = \frac{1 + 65}{6} = \frac{66}{6} = 11
\]

or

\[
x = \frac{1 - 65}{6} = \frac{-64}{6} = -\frac{32}{3}
\]

Since the number of rows must be a positive integer, we discard the negative solution. Thus, the number of rows is \( x = 11 \).

Now, the number of students in each row is:

\[
3x - 1 = 3(11) - 1 = 33 - 1 = 32
\]

Therefore, the number of students in each row is **32**.

Would you like further details or explanations? Here are some related questions:

1. How is the quadratic formula derived?
2. How do you determine when to discard a solution in a quadratic equation?
3. Can you explain how to complete the square for this quadratic equation?
4. What are the alternative methods to solve quadratic equations?
5. How do you interpret the discriminant in this context?

**Tip:** Always check for extraneous solutions, especially when working with word problems involving real-world constraints like positive integers.

A class of 352 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.

Learn how to solve a quadratic equation in this word problem involving 352 students arranged in rows. This detailed step-by-step solution covers algebraic concepts, the quadratic formula, and real-world problem-solving. Perfect for students in Grades 8-10.

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