Let's solve the problem step-by-step to find the two consecutive odd integers whose product is 99.
1. Define the integers:
Let the first odd integer be [tex]\( x \)[/tex].
The next consecutive odd integer would then be [tex]\( x + 2 \)[/tex].
2. Set up the equation:
According to the problem, the product of these two integers is 99. This can be written as:
[tex]\[
x \times (x + 2) = 99
\][/tex]
3. Form a quadratic equation:
Expanding the product, we get:
[tex]\[
x^2 + 2x = 99
\][/tex]
Rearrange this to form a standard quadratic equation:
[tex]\[
x^2 + 2x - 99 = 0
\][/tex]
4. Solve the quadratic equation:
To find the values of [tex]\( x \)[/tex], we use the quadratic formula:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -99 \)[/tex].
5. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-99) = 4 + 396 = 400
\][/tex]
6. Find the roots:
[tex]\[
x = \frac{{-2 \pm \sqrt{400}}}{2} = \frac{{-2 \pm 20}}{2}
\][/tex]
This gives us two solutions:
[tex]\[
x_1 = \frac{{-2 + 20}}{2} = \frac{18}{2} = 9
\][/tex]
[tex]\[
x_2 = \frac{{-2 - 20}}{2} = \frac{-22}{2} = -11
\][/tex]
7. Select the positive integer:
Since we are asked to provide the positive integers only, we choose [tex]\( x = 9 \)[/tex].
8. Find the next consecutive odd integer:
The next consecutive odd integer is [tex]\( x + 2 = 9 + 2 = 11 \)[/tex].
Therefore, the two consecutive odd integers whose product is 99 are [tex]\( \boxed{9 \text{ and } 11} \)[/tex].
