two positive, consecutive, odd integers have a product of 143. complete the equation to represent finding x…

Explanation:

Step1: Define consecutive odd - integers

Let the greater odd integer be $x$. Then the smaller consecutive odd integer is $x - 2$ (since the difference between consecutive odd integers is 2).

Step2: Form the equation

The product of the two odd integers is 143. So the equation is $x(x - 2)=143$.

Step3: Expand and solve the quadratic equation

Expand $x(x - 2)=143$ to get $x^{2}-2x - 143 = 0$. Factor the quadratic equation: $x^{2}-2x - 143=(x - 13)(x + 11)=0$. Set each factor equal to zero: $x-13 = 0$ gives $x = 13$; $x + 11=0$ gives $x=-11$. Since we are looking for positive odd integers, we discard $x=-11$.

Answer:

The number to fill in the blank is 2, and the greater integer is 13.