Question

the sum of squares of 2 +ve integers is 106 .sum of larger integer and thrice small integer is 24 . find the nos (use only 1 variable)

Answer 1

$\large \underline{\tt \: \red{ According \: to \: statement }}$

• Sum of larger integer and thrice small integer is 24.

So,

• Let smaller integer = 'x'.

So,

• Largest integer = 24 - 3x

$\large \underline{\tt \: \blue{ According \: to \: statement }}$

$\tt \longmapsto\: {x}^{2} + {(24 - 3x)}^{2} = 106$

$\tt \longmapsto\: {x}^{2} + 576 + {9x}^{2} - 144x = 106$

$\tt \longmapsto\:10 {x}^{2} - 144x + 470 = 0$

$\tt \longmapsto\: 5{x}^{2} - 72x + 235 = 0$

$\tt \longmapsto\:5 {x}^{2} - 25x - 47x + 235 = 0$

$\tt \longmapsto\:5x(x - 5) - 47(x - 5) = 0$

$\tt \longmapsto\:(x - 5)(5x - 47) = 0$

$\bf\implies \:x = 5 \: \: or \: x \: = \dfrac{47}{5} (rejected)$

$\rm :\implies\:\:\boxed{ \green{ \bf \: x = 5}}$

So,

$\begin{gathered}\begin{gathered}\bf \:numbers \: are - \begin{cases} &\tt{x = 5} \\ &\tt{24 - 3x = 24 - 15 = 9} \end{cases}\end{gathered}\end{gathered}$