Given:
The sum of the squares of two consecutive odd integers is 514.
Required:
Find the integers.
Explanation:
Let two consecutive odd integers be x, x+2.
According to the question
[tex]\begin{gathered} x^2+(x+2)^2=514 \\ x^2+x^2+4x+4=514 \\ 2x^2+4x-510=0 \\ x^2+2x-255=0 \end{gathered}[/tex]This is the quadratic equation.
Solve it by using the middle-term splitting method.
[tex]\begin{gathered} x^2+17x-15x-255=0 \\ x(x+17)-15(x-17)=0 \\ (x+17)(x-15)=0 \end{gathered}[/tex][tex]\begin{gathered} x+17=0 \\ x=-17 \end{gathered}[/tex][tex]\begin{gathered} x-15=0 \\ x=15 \end{gathered}[/tex]When x=-17 then consecutive odd integer = -17+5 = -15
When x= 15 then consecutive odd integer =15+ 2 = 17
Final Answer:
The consecutive odd integer
