Answer :
Correct Question : The sum of squares of two consecutive positive odd numbers is 514. Find the numbers.
[tex]\rule{130}1[/tex]
Solution :
Let the two consecutive positive odd numbers be x and (x + 2)
[tex]\rule{130}1[/tex]
☯ [tex]\underline{\boldsymbol{According\: to \:the\: Question\:now :}}[/tex]
[tex]:\implies\sf x^2 + (x + 2)^{2} = 514 \\\\\\:\implies\sf x^2 + x^2 + 4x + 4 = 514\\\\\\:\implies\sf 2x^2 + 4x + 4 = 514 \\\\\\:\implies\sf 2x^2 + 4x = 514 - 4\\\\\\:\implies\sf 2x^2 + 4x = 510 \\\\\\:\implies\sf 2x^2 + 4x - 510 = 0\\\\\\:\implies\sf x^2 + 2x - 255 = 0 \:\:\:\:\:\Bigg\lgroup \bf{Dividing\:by\:2}\Bigg\rgroup\\\\\\:\implies\sf x^2 + 17x - 15x - 255 = 0\\\\\\:\implies\sf x(x + 17) - 15(x + 17) = 0\\\\\\:\implies\sf (x + 17) (x - 15)\\\\\\:\implies\sf x = - 17\:or\:x = 15\\\\\\:\implies\underline{\boxed {\sf x = 15}}\:\:\:\:\:\Bigg\lgroup \bf{Positive\:odd\: number}\Bigg\rgroup[/tex]
When,
[tex]\bullet\: \textsf {x = } \textbf {15} [/tex]
[tex]\bullet\: \sf x + 2 = 15 + 2 =\textbf{ 17} [/tex]
[tex]\therefore\:\underline{\textsf{The required positive integer is \textbf{15 and 17}}}.[/tex]