The product of two consecutive odd integers is 675. What is the smaller integer?

The correct answer is 25.

Step 1: Define the Variables
Let the smaller odd integer be represented by $x$.

Since consecutive odd integers differ by 2, the next consecutive odd integer is represented by $x+2$.

Step 2: Set Up the Product Equation
We are given that the product of these two consecutive odd integers is 675. Therefore, we can set up the equation:

$x(x+2)=675$

Step 3: Expand and Rearrange into Standard Quadratic Form
Distribute $x$ on the left side of the equation:

$x^2+2x=675$

Subtract 675 from both sides to form a quadratic equation equal to zero:

$x^2+2x-675=0$

Step 4: Solve the Quadratic Equation by Factoring
To factor $x^2+2x-675$, we look for two numbers whose product is -675 and whose sum is +2.
Testing factors near $\sqrt{675}\approx 25.98$, we find that:

$27\times (-25)=-675$

$27+(-25)=2$

Rewriting the quadratic equation in factored form:

$(x-25)(x+27)=0$

Set each factor equal to zero to find the possible values for $x$:

$x-25=0\Rightarrow x=25$

$x+27=0\Rightarrow x=-27$

Step 5: Verification
Considering the positive integer solution, the smaller integer is $x=25$ and the larger consecutive odd integer is $x+2=27$.

Checking their product:

$25\times 27=675$

Thus, the smaller integer is 25.