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$4000k = 4000 \left( 1+ \dfrac{ 5x }{ 100 } \right) \times k \left( 1- \dfrac{ 4x }{ 100 } \right)$
$k < 0 $ or $ k > 0 , $ 2 real roots $ $
Find the number of solutions
$4000 k = \color{#FF6800}{ 4000 } \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 5 x } { 100 } } \right ) \color{#FF6800}{ k } \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 4 x } { 100 } } \right )$
$ $ Move the expression to the left side and change the symbol $ $
$4000 k - 4000 \left ( 1 + \dfrac { 5 x } { 100 } \right ) k \left ( 1 - \dfrac { 4 x } { 100 } \right ) = 0$
$\color{#FF6800}{ 4000 } \color{#FF6800}{ k } \color{#FF6800}{ - } \color{#FF6800}{ 4000 } \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 5 x } { 100 } } \right ) \color{#FF6800}{ k } \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 4 x } { 100 } } \right ) = \color{#FF6800}{ 0 }$
$ $ When it's $ k = 0 $ , the given equation is not quadratic, so discriminant cannot be used $ $
$\color{#FF6800}{ 4000 } \color{#FF6800}{ k } - 4000 \left ( 1 + \dfrac { 5 x } { 100 } \right ) k \left ( 1 - \dfrac { 4 x } { 100 } \right ) = 0 \left ( \text{However (or only)} \color{#FF6800}{ k } \neq \color{#FF6800}{ 0 } \right )$
$\color{#FF6800}{ 4000 } \color{#FF6800}{ k } \color{#FF6800}{ - } \color{#FF6800}{ 4000 } \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { 5 x } { 100 } } \right ) \color{#FF6800}{ k } \left ( \color{#FF6800}{ 1 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 4 x } { 100 } } \right ) = \color{#FF6800}{ 0 }$
$ $ In the quadratic equation $ ax^{2}+bx+c=0 $ , use the discriminant $ D=b^{2}-4ac $ to determine the number of solutions $ $
$\color{#FF6800}{ D } = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 40 } \color{#FF6800}{ k } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \left ( \color{#FF6800}{ 8 } \color{#FF6800}{ k } \right ) \color{#FF6800}{ \times } \color{#FF6800}{ 0 }$
$D = \left ( \color{#FF6800}{ - } \color{#FF6800}{ 40 } \color{#FF6800}{ k } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \left ( \color{#FF6800}{ 8 } \color{#FF6800}{ k } \right ) \color{#FF6800}{ \times } \color{#FF6800}{ 0 }$
$ $ Organize the expression $ $
$D = \color{#FF6800}{ 1600 } \color{#FF6800}{ k } ^ { \color{#FF6800}{ 2 } }$
$\color{#FF6800}{ D } = \color{#FF6800}{ 1600 } \color{#FF6800}{ k } ^ { \color{#FF6800}{ 2 } }$
$ $ The number of solutions depends on the range of the discriminant $ D$
$\color{#FF6800}{ 1600 } \color{#FF6800}{ k } ^ { \color{#FF6800}{ 2 } } > \color{#FF6800}{ 0 } , $ 2 real roots $ \\ \color{#FF6800}{ 1600 } \color{#FF6800}{ k } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 } , $ One real root $ \\ \color{#FF6800}{ 1600 } \color{#FF6800}{ k } ^ { \color{#FF6800}{ 2 } } < \color{#FF6800}{ 0 } , $ Do not have the real root $ $
$\color{#FF6800}{ 1600 } \color{#FF6800}{ k } ^ { \color{#FF6800}{ 2 } } > \color{#FF6800}{ 0 } , $ 2 real roots $ \\ 1600 k ^ { 2 } = 0 , $ One real root $ \\ 1600 k ^ { 2 } < 0 , $ Do not have the real root $ $
$ $ Solve a solution to $ k$
$\color{#FF6800}{ k } < \color{#FF6800}{ 0 } $ or $ \color{#FF6800}{ k } > \color{#FF6800}{ 0 } , $ 2 real roots $ \\ 1600 k ^ { 2 } = 0 , $ One real root $ \\ 1600 k ^ { 2 } < 0 , $ Do not have the real root $ $
$k < 0 $ or $ k > 0 , $ 2 real roots $ \\ \color{#FF6800}{ 1600 } \color{#FF6800}{ k } ^ { \color{#FF6800}{ 2 } } = \color{#FF6800}{ 0 } , $ One real root $ \\ 1600 k ^ { 2 } < 0 , $ Do not have the real root $ $
$ $ Find a solution for the equation $ $
$k < 0 $ or $ k > 0 , $ 2 real roots $ \\ \color{#FF6800}{ k } = \color{#FF6800}{ 0 } , $ One real root $ \\ 1600 k ^ { 2 } < 0 , $ Do not have the real root $ $
$k < 0 $ or $ k > 0 , $ 2 real roots $ \\ k = 0 , $ One real root $ \\ \color{#FF6800}{ 1600 } \color{#FF6800}{ k } ^ { \color{#FF6800}{ 2 } } < \color{#FF6800}{ 0 } , $ Do not have the real root $ $
$ $ Solve a solution to $ k$
$k < 0 $ or $ k > 0 , $ 2 real roots $ \\ k = 0 , $ One real root $ \\ \color{#FF6800}{ k } \in \emptyset \left ( \text{Do not have the solution} \right ) , $ Do not have the real root $ $
$\color{#FF6800}{ k } < \color{#FF6800}{ 0 } $ or $ \color{#FF6800}{ k } > \color{#FF6800}{ 0 } , $ 2 real roots $ \\ \color{#FF6800}{ k } = \color{#FF6800}{ 0 } , $ One real root $ \\ \color{#FF6800}{ k } \in \emptyset \left ( \text{Do not have the solution} \right ) , $ Do not have the real root $ $
$ $ Rewrite the answer $ $
$\color{#FF6800}{ k } < \color{#FF6800}{ 0 } $ or $ \color{#FF6800}{ k } > \color{#FF6800}{ 0 } , $ 2 real roots $ \\ \color{#FF6800}{ k } = \color{#FF6800}{ 0 } , $ One real root $ $
$k < 0 $ or $ k > 0 , $ 2 real roots $ \\ \color{#FF6800}{ k } = \color{#FF6800}{ 0 } , $ One real root $ $
$ $ Exclude the range of $ k \neq 0 $ where the previously calculated discriminant cannot be used $ $
$k < 0 $ or $ k > 0 , $ 2 real roots $ \\ \color{#FF6800}{ k } \in \emptyset \left ( \text{Do not have the solution} \right ) , $ One real root $ $
$\color{#FF6800}{ k } < \color{#FF6800}{ 0 } $ or $ \color{#FF6800}{ k } > \color{#FF6800}{ 0 } , $ 2 real roots $ \\ \color{#FF6800}{ k } \in \emptyset \left ( \text{Do not have the solution} \right ) , $ One real root $ $
$ $ Rewrite the answer $ $
$\color{#FF6800}{ k } < \color{#FF6800}{ 0 } $ or $ \color{#FF6800}{ k } > \color{#FF6800}{ 0 } , $ 2 real roots $ $
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