Question

X/3 + 3/x = 17/4 find the quadratic equations.

Answer 1

$\red {\huge{ \underline{ \overline{ \mid{ \bold {\purple{ \: \:SOLUTION \: \: \red{ \mid}}}}}}}}$

$\bold{ \: \: \: \: \: \: \: \: \: \frac{x}{3} + \frac{3}{x} = \frac{17}{4} } \\ \\ \bold{ \Longrightarrow \: \frac{x {}^{2} + 9 }{3x} = \frac{17}{4} } \\ \\ \bold{ \Longrightarrow \: 4(x {}^{2} + 9) = 17 \times 3x } \\ \\ \bold{ \Longrightarrow \: 4x {}^{2} + 36 = 51x} \\ \\ \bold{ \Longrightarrow \: 4x {}^{2} - 51x + 36= 0}$

$\underline{ \bold{ \: \: The \: \: quadratic \: \: equation \: \: is \: : }} \\ \\ \: \: \: \: \: \: \: \: \: \boxed{ \boxed{ \bold{ \red{ \: \: 4x {}^{2} - 51x + 36 = 0 \: \: }}}}$

$\underline{ \bold{ \: \: We \: \: know \: \: that \: \: }}$

$\bold{In \: \: a \: \: quadratic \: \: equation \: \: which \: \: is \: \: } \\ \bold{in \: \: the \: \: form \: \: of \: \: ax {}^{2} + bx + c = 0 } \\ \\ \bold{the \: \: roots \: \: of \: \: the \: \: equation \: \: is \: \: find \: } \\ \bold{by \: \: the \: \:quadratic \: \: formula \: : - }$

$\boxed{ \bold{ \: \: \: x = \frac{ \: - \: b \: \pm \: \sqrt{b {}^{2} \: - \: 4ac \: \: } \: \: }{2a} \: \: }}$

$\bold{Here,} \\ \\ \bold{a = 4} \\ \\ \bold{b = - 51} \\ \\ \bold{c = 36}$

$\underline{ \bold{ \: \: Using \: \: the \: \: quadratic \: \: formula \: \: }}$

$\bold{x = \frac{ - b \: \pm \: \sqrt{b {}^{2} - 4ac \: } \: }{2a} } \\ \\ \bold{ = \frac{ - ( - 51) \: \pm \: \sqrt{( - 51) {}^{2} \: - \: 4 \times 4 \times 36 \: } \: }{2 \times 4} } \\ \\ \bold{ = \frac{ 51 \: \pm \: \sqrt{2601 - 576 \: } \: \: }{8} } \\ \\ \bold{ = \frac{ 51 \: \pm \: \sqrt{2025 \: } }{8} } \\ \\ \bold{ = \frac{ 51 \: \pm \: 45}{8} }$

$\bold{ \therefore \: \: x = \frac{ \: \: 51 \: - \: 45 \: }{8} } \\ \\ \bold{ \Longrightarrow \: x = \frac{ \: 6 \: \: }{8} } \\ \\ \bold{ \Longrightarrow \: x = \frac{ 3 \: }{4} }$

$\bold{Or,} \\ \\ \bold{x = \frac{ 51 + 45}{8} } \\ \\ \bold{ \Longrightarrow \: x = \frac{ 96 \: }{8} } \\ \\ \bold{ \Longrightarrow \: x = 12}$

$\bold{\therefore \: \: The \: \: roots \: \: of \: \: the \: \: quadratic \: \: equation \: \: is \: : -} \\ \\ \bold{\red{ \: x \: = \: \frac{ \: 3}{4} \: \: \: \: Or , \: \: \: \: x \: = \: \: 12 \: }}$