Math Problem Statement

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.

Solution

Let the number of rows be xx. The number of students in each row is then 3x13x - 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×(3x1)=352x \times (3x - 1) = 352

Expanding the equation:

3x2x=3523x^2 - x = 352

Rearranging the equation to standard quadratic form:

3x2x352=03x^2 - x - 352 = 0

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

x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

For the equation 3x2x352=03x^2 - x - 352 = 0, the coefficients are:

  • a=3a = 3
  • b=1b = -1
  • c=352c = -352

Substituting into the quadratic formula:

x=(1)±(1)24(3)(352)2(3)x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-352)}}{2(3)}

x=1±1+42246x = \frac{1 \pm \sqrt{1 + 4224}}{6}

x=1±42256x = \frac{1 \pm \sqrt{4225}}{6}

x=1±656x = \frac{1 \pm 65}{6}

This gives two possible solutions:

x=1+656=666=11x = \frac{1 + 65}{6} = \frac{66}{6} = 11

or

x=1656=646=323x = \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=11x = 11.

Now, the number of students in each row is:

3x1=3(11)1=331=323x - 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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Word Problems

Formulas

Quadratic equation formula: ax^2 + bx + c = 0
Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a

Theorems

Quadratic formula
Solving quadratic equations in real-world contexts

Suitable Grade Level

Grades 8-10