I was solving a puzzle, and finally I got this equation as the result but I couldn't solve it.
$$x+\left(x\times\left(\frac{20}{100}\right)\right)=600$$
How to solve this equation?
Please provide a step by step solution.
[Math] How to solve $x+(x\cdot\frac{20}{100})=600$
algebra-precalculus
Related Solutions
There is a quite dirty trick you can use here. Define $$ f(x,t)=(x^2+t)^2-(x^3-t)^3\ . $$ Then solve $f(x,t)=0$ for $t$ (not for $x$). There is a simple solution of the cubic equation in $t$ $$ t=x^2(x-1)\ . $$ Then set $t=100$ and solve the cubic equation for $x$, yielding $x=5$ as the only real root.
$$e^{\frac{x^2}{4vt}} = 1+\frac{x^2}{2vt}$$ Let $y=\frac{x^2}{4vt}$ $$e^y=1+2y$$ $$e^{-y}=\frac{1}{1+2y}$$ $$(1+2y)e^{-y}=1$$ $$(\frac12+y)e^{-y}=\frac12$$ $$(-\frac12-y)e^{-y}=-\frac12$$ $$ (-\frac12-y)e^{-y}e^{-\frac12}=-\frac12 e^{-\frac12}$$ $$ (-\frac12-y)e^{-\frac12-y}=-\frac12 e^{-\frac12}=-\frac{1}{2\sqrt{e}}$$ $X=(-\frac12-y)$ $$Xe^X=-\frac{1}{2\sqrt{e}}$$ From the definition of the Lambert W function : http://mathworld.wolfram.com/LambertW-Function.html $$X=W\left(-\frac{1}{2\sqrt{e}}\right)$$ $$y=-\frac12-X=-\frac12-W\left(-\frac{1}{2\sqrt{e}}\right)$$ $$x=\sqrt{4vty}=\sqrt{-2-4W\left(-\frac{1}{2\sqrt{e}}\right)}\sqrt{vt}$$ The Lambert W(z) function is a multi valuated function in $-\frac{1}{e} <z<0$ , real $z$.
This is presently the case where $z=-\frac{1}{2\sqrt{e}}$ since $-\frac{1}{e} <-\frac{1}{2\sqrt{e}}<0$
First root :
$W_0\left(-\frac{1}{2\sqrt{e}}\right)=-\frac12 \quad;\quad {-2-4W_0\left(-\frac{1}{2\sqrt{e}}\right)}=-2-4(-1/2)=0 \quad;\quad x=0$
Second root :
$W_{-1}\left(-\frac{1}{2\sqrt{e}}\right)\simeq -1.756431... $
https://www.wolframalpha.com/input/?i=lambertw(-1,-1%2F(2+sqrt(e)))
One can use series expansion to compute it approximately. See page 13 in https://fr.scribd.com/doc/34977341/Sophomore-s-Dream-Function . The convergence is very slow. See the numerical calculus in the below addition. In practice, it is certainly faster to solve directly $e^y=1+2y$ with Newton-Raphson or similar iterative method.
Finally, an approximate value is :
$$x\simeq\sqrt{-2-4(-1.756431)}\sqrt{vt}\simeq 2.2418128 \sqrt{vt}$$
With more digits : https://www.wolframalpha.com/input/?i=sqrt(-2-4+lambertw(-1,-1%2F(2+sqrt(e))))
IN ADDITION :
Example of recursive numerical calculus of $W_{-1}(x)$ in the range $-\frac{1}{e}<x<0$
Best Answer
$$x+\left(x\times\left(\frac{20}{100}\right)\right)=600$$
$$x+\left(x\times\left(\frac{1}{5}\right)\right)=600$$
$$x+ \frac{x}{5}=600$$
$$5x+ {x}=600 * 5$$
$$6x=3000$$
$$x=500$$