Solve -t^2+2t+t^2-t/2=k(t-t^2-t/2)

$$- t ^ { 2 } + 2 t + \frac { t ^ { 2 } - t } { 2 } = k ( t - \frac { t ^ { 2 } - t } { 2 } )$$

Solve for k

$$\left\{\begin{matrix}\\k=1\text{, }&\text{unconditionally}\\k\in \mathrm{R}\text{, }&t=3\text{ or }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$-2t^{2}+4t+t^{2}-t=2k\left(t-\frac{t^{2}-t}{2}\right)$$

Combine -2t^{2} and t^{2} to get -t^{2}.

$$-t^{2}+4t-t=2k\left(t-\frac{t^{2}-t}{2}\right)$$

Combine 4t and -t to get 3t.

$$-t^{2}+3t=2k\left(t-\frac{t^{2}-t}{2}\right)$$

Use the distributive property to multiply 2k by t-\frac{t^{2}-t}{2}.

$$-t^{2}+3t=2kt+2k\left(-\frac{t^{2}-t}{2}\right)$$

Express 2\left(-\frac{t^{2}-t}{2}\right) as a single fraction.

$$-t^{2}+3t=2kt+\frac{-2\left(t^{2}-t\right)}{2}k$$

Cancel out 2 and 2.

$$-t^{2}+3t=2kt-\left(t^{2}-t\right)k$$

Use the distributive property to multiply -1 by t^{2}-t.

$$-t^{2}+3t=2kt+\left(-t^{2}+t\right)k$$

Use the distributive property to multiply -t^{2}+t by k.

$$-t^{2}+3t=2kt-t^{2}k+tk$$

Combine 2kt and tk to get 3kt.

$$-t^{2}+3t=3kt-t^{2}k$$

Swap sides so that all variable terms are on the left hand side.

$$3kt-t^{2}k=-t^{2}+3t$$

Combine all terms containing k.

$$\left(3t-t^{2}\right)k=-t^{2}+3t$$

The equation is in standard form.

$$\left(3t-t^{2}\right)k=3t-t^{2}$$

Divide both sides by 3t-t^{2}.

$$\frac{\left(3t-t^{2}\right)k}{3t-t^{2}}=\frac{t\left(3-t\right)}{3t-t^{2}}$$

Dividing by 3t-t^{2} undoes the multiplication by 3t-t^{2}.

$$k=\frac{t\left(3-t\right)}{3t-t^{2}}$$

Divide t\left(3-t\right) by 3t-t^{2}.

$$k=1$$

Solve for t

$$\left\{\begin{matrix}\\t=3\text{; }t=0\text{, }&\text{unconditionally}\\t\in \mathrm{R}\text{, }&k=1\end{matrix}\right.$$