Respuesta :
SOLUTION
Given the question in the question tab, the following are the solution steps for getting the answers
Step 1: Let the odd integers be 2n+1 and 2n+3, where n is an integer. The statements in the question can be written as:
[tex]2n+1(2n+3)=83+4(2n+1+2n+3)[/tex]Step 2: Solve tne mathematical equation to get the value of n
[tex]\begin{gathered} \text{ Removing the bracket} \\ 4n^2+6n+2n+3=83+4(4n+4)_{} \\ 4n^2+8n+3=83+16n+16 \\ 4n^2+8n+3=16n+99 \\ 4n^2+8n+3-16n-99 \\ 4n^2-8n-96 \\ \text{Divide through by 2, we have} \\ n^2-2n-24 \end{gathered}[/tex]Step 3: Solve the quadratic equation in step 2 using formular method to get the values of n
[tex]\begin{gathered} a=1,b=-2,c=-24 \\ \text{formula}=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \Rightarrow\frac{-(-2)\pm\sqrt[]{(-2)^2-4(1)(-24)}}{2(1)}=\frac{2\pm\sqrt[]{4+96}_{}}{2}=\frac{2\pm\sqrt[]{100}}{2} \\ \Rightarrow\frac{2+10}{2},\frac{2-10}{2} \\ \Rightarrow\frac{12}{2},-\frac{8}{2} \\ \Rightarrow6,-4 \end{gathered}[/tex]Step 4: Substitute the values gotten for n in the expression given in step 1 to get the odd numbers
[tex]\begin{gathered} \text{when n=6} \\ 2n+1=2(6)+1=12+1=13 \\ 2n+3=2(6)+3=12+3=15 \\ \text{when n=-4} \\ 2n+1=2(-4)+1=-8+1=-7 \\ 2n+3=2(-4)+3=-8+3=-5 \end{gathered}[/tex]It can be seen from the solution above that there are two solutions.
[tex]\begin{gathered} \text{The smaller consecutive odd integers are }(-7,-5) \\ \text{The larger consecutive odd integers are }(13,15) \end{gathered}[/tex]