Math Problem Statement

two consecutive odd numbers are such that their product is 35.find the numbers

Solution

Let's define the two consecutive odd numbers as xx and x+2x + 2, where xx is the first odd number, and x+2x + 2 is the next consecutive odd number.

We are given that their product is 35, so we can set up the equation:

x(x+2)=35x(x + 2) = 35

Now, expand the left side:

x2+2x=35x^2 + 2x = 35

Next, subtract 35 from both sides to set the equation to 0:

x2+2x35=0x^2 + 2x - 35 = 0

Now, we can solve this quadratic equation. We can either use factoring, completing the square, or the quadratic formula. Let's try factoring:

We need two numbers that multiply to 35-35 and add up to 22. These numbers are 77 and 5-5. So, we can factor the equation as:

(x+7)(x5)=0(x + 7)(x - 5) = 0

Now, set each factor equal to 0:

x+7=0orx5=0x + 7 = 0 \quad \text{or} \quad x - 5 = 0

Solving these gives:

x=7orx=5x = -7 \quad \text{or} \quad x = 5

Thus, the two possible pairs of consecutive odd numbers are:

  1. If x=7x = -7, the numbers are 7-7 and 5-5.
  2. If x=5x = 5, the numbers are 55 and 77.

Conclusion:

The two consecutive odd numbers are either 7-7 and 5-5, or 55 and 77.

Would you like further clarification or details on this process?


Here are 5 related questions:

  1. How do you solve a quadratic equation using the quadratic formula?
  2. What are other methods for solving quadratic equations besides factoring?
  3. How do you determine if a number is odd or even?
  4. What is the significance of factoring when solving quadratic equations?
  5. How do you verify the solutions to quadratic equations?

Tip: When solving quadratic equations, always check the factored form to make sure both roots satisfy the original equation!

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations

Formulas

Quadratic equation formula ax^2 + bx + c = 0

Theorems

Factoring quadratic equations

Suitable Grade Level

Grades 7-9