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Formula
Number of solution
$4000k = 4000 \left( 1+ \dfrac{ 5x }{ 100 } \right) \times k \left( 1- \dfrac{ 4x }{ 100 } \right)$
$k < 0$ 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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