Solve k*50*5/x^2=k*30*5/left(80-xright)^2

$$\frac{ k \times 50 \times 5 }{ { x }^{ 2 } } = \frac{ k \times 30 \times 5 }{ { \left(80-x \right) }^{ 2 } }$$

Solve for k (complex solution)

$$\left\{\begin{matrix}k=0\text{, }&x\neq 0\text{ and }x\neq 80\\k\in \mathrm{C}\text{, }&x=200-40\sqrt{15}\text{ or }x=40\sqrt{15}+200\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}\left(x-80\right)^{2}, the least common multiple of x^{2},\left(80-x\right)^{2}.

$$\left(x-80\right)^{2}k\times 50\times 5=x^{2}k\times 30\times 5$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-80\right)^{2}.

$$\left(x^{2}-160x+6400\right)k\times 50\times 5=x^{2}k\times 30\times 5$$

Multiply 50 and 5 to get 250.

$$\left(x^{2}-160x+6400\right)k\times 250=x^{2}k\times 30\times 5$$

Use the distributive property to multiply x^{2}-160x+6400 by k.

$$\left(x^{2}k-160xk+6400k\right)\times 250=x^{2}k\times 30\times 5$$

Use the distributive property to multiply x^{2}k-160xk+6400k by 250.

$$250x^{2}k-40000xk+1600000k=x^{2}k\times 30\times 5$$

Multiply 30 and 5 to get 150.

$$250x^{2}k-40000xk+1600000k=x^{2}k\times 150$$

Subtract x^{2}k\times 150 from both sides.

$$250x^{2}k-40000xk+1600000k-x^{2}k\times 150=0$$

Combine 250x^{2}k and -x^{2}k\times 150 to get 100x^{2}k.

$$100x^{2}k-40000xk+1600000k=0$$

Combine all terms containing k.

$$\left(100x^{2}-40000x+1600000\right)k=0$$

Divide 0 by 100x^{2}-40000x+1600000.

$$k=0$$

Solve for k

$$\left\{\begin{matrix}k=0\text{, }&x\neq 0\text{ and }x\neq 80\\k\in \mathrm{R}\text{, }&x=40\sqrt{15}+200\text{ or }x=200-40\sqrt{15}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}\left(x-80\right)^{2}, the least common multiple of x^{2},\left(80-x\right)^{2}.

$$\left(x-80\right)^{2}k\times 50\times 5=x^{2}k\times 30\times 5$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-80\right)^{2}.

$$\left(x^{2}-160x+6400\right)k\times 50\times 5=x^{2}k\times 30\times 5$$

Multiply 50 and 5 to get 250.

$$\left(x^{2}-160x+6400\right)k\times 250=x^{2}k\times 30\times 5$$

Use the distributive property to multiply x^{2}-160x+6400 by k.

$$\left(x^{2}k-160xk+6400k\right)\times 250=x^{2}k\times 30\times 5$$

Use the distributive property to multiply x^{2}k-160xk+6400k by 250.

$$250x^{2}k-40000xk+1600000k=x^{2}k\times 30\times 5$$

Multiply 30 and 5 to get 150.

$$250x^{2}k-40000xk+1600000k=x^{2}k\times 150$$

Subtract x^{2}k\times 150 from both sides.

$$250x^{2}k-40000xk+1600000k-x^{2}k\times 150=0$$

Combine 250x^{2}k and -x^{2}k\times 150 to get 100x^{2}k.

$$100x^{2}k-40000xk+1600000k=0$$

Combine all terms containing k.

$$\left(100x^{2}-40000x+1600000\right)k=0$$

Divide 0 by 100x^{2}-40000x+1600000.

$$k=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=40\sqrt{15}+200\text{; }x=200-40\sqrt{15}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\setminus 0,80\text{, }&k=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=40\sqrt{15}+200\text{; }x=200-40\sqrt{15}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\setminus 0,80\text{, }&k=0\end{matrix}\right.$$