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

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

$x(x + 2) = 35$

Now, expand the left side:

$x^2 + 2x = 35$

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

$x^2 + 2x - 35 = 0$

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 $-35$ and add up to $2$. These numbers are $7$ and $-5$. So, we can factor the equation as:

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

Now, set each factor equal to 0:

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

Solving these gives:

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

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

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

Conclusion:

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

Would you like further clarification or details on this process?

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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!