How do you solve x(1-x)+2x-4=8x-24-x^2?

Jan 6, 2017

See full process in the Explanation

Explanation:

First step is to expand the terms within the parenthesis:

$x \left(1 - x\right) + 2 x - 4 = 8 x - 24 - {x}^{2}$

$x - {x}^{2} + 2 x - 4 = 8 x - 24 - {x}^{2}$

We can now group and combine like terms:

$x + 2 x - {x}^{2} - 4 = 8 x - 24 - {x}^{2}$

$3 x - {x}^{2} - 4 = 8 x - 24 - {x}^{2}$

We can now add and subtract the necessary terms to isolate the $x$ terms while keeping the equation balanced:

$3 x - {x}^{2} - 4 - \textcolor{red}{3 x} + \textcolor{b l u e}{{x}^{2}} + \textcolor{g r e e n}{24} = 8 x - 24 - {x}^{2} - \textcolor{red}{3 x} + \textcolor{b l u e}{{x}^{2}} + \textcolor{g r e e n}{24}$

$3 x - \textcolor{red}{3 x} - {x}^{2} + \textcolor{b l u e}{{x}^{2}} - 4 + \textcolor{g r e e n}{24} = 8 x - \textcolor{red}{3 x} - 24 + \textcolor{g r e e n}{24} - {x}^{2} + \textcolor{b l u e}{{x}^{2}}$

$0 - 0 - 4 + \textcolor{g r e e n}{24} = 8 x - \textcolor{red}{3 x} - 0 - 0$

$- 4 + \textcolor{g r e e n}{24} = 8 x - \textcolor{red}{3 x}$

$20 = 5 x$

Now we can divide each side of the equation by $\textcolor{red}{5}$ to solve for $x$ while keeping the equation balanced:

$\frac{20}{\textcolor{red}{5}} = \frac{5 x}{\textcolor{red}{5}}$

$4 = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{5}}} x}{\cancel{\textcolor{red}{5}}}$

$4 = x$

$x = 4$