To solve this problem, we need to find two positive consecutive odd integers whose product is 143.
Let's denote the greater odd integer by [tex]\( x \)[/tex]. Since the integers are odd and consecutive, the smaller odd integer would then be [tex]\( x - 2 \)[/tex].
The product of these two integers can be represented by the equation:
[tex]\[ x(x - 2) = 143 \][/tex]
Now, let's set up the equation:
[tex]\[ x(x - 2) = 143 \][/tex]
Expanding this, we get:
[tex]\[ x^2 - 2x = 143 \][/tex]
Rearranging it into standard form of a quadratic equation:
[tex]\[ x^2 - 2x - 143 = 0 \][/tex]
To solve this quadratic equation, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -143 \)[/tex].
Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-2)^2 - 4 \cdot 1 \cdot (-143) \][/tex]
[tex]\[ \Delta = 4 + 572 \][/tex]
[tex]\[ \Delta = 576 \][/tex]
Now, find the roots:
[tex]\[ x = \frac{2 \pm \sqrt{576}}{2} \][/tex]
[tex]\[ x = \frac{2 \pm 24}{2} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{26}{2} = 13 \][/tex]
[tex]\[ x_2 = \frac{-22}{2} = -11 \][/tex]
Since we are looking for positive integers, we discard [tex]\( -11 \)[/tex] and accept [tex]\( 13 \)[/tex] as the solution.
Thus, the greater integer is:
[tex]\[ \boxed{13} \][/tex]
