To find two positive, consecutive, odd integers whose product is 143, we can follow this step-by-step method.
Let the greater integer be [tex]\( x \)[/tex]. Since we are dealing with consecutive odd integers, the smaller integer will then be [tex]\( x - 2 \)[/tex].
The equation representing their product can be written as:
[tex]\[ x(x - 2) = 143 \][/tex]
We are asked to solve for [tex]\( x \)[/tex], the greater integer. The equation to solve is a quadratic equation:
[tex]\[ x^2 - 2x = 143 \][/tex]
Rewriting the equation, we get:
[tex]\[ x^2 - 2x - 143 = 0 \][/tex]
Solving this quadratic equation, we find that:
[tex]\[ x = 13 \][/tex]
Thus, the two positive, consecutive, odd integers are [tex]\( 13 \)[/tex] and [tex]\( 11 \)[/tex].
The greater integer is:
[tex]\[ \boxed{13} \][/tex]
