Solve (-2ce^{-2(3)}-e^-(3)+2(ce^-2(3)+e^-(3))=e^-(3)

$$( - 2 c e ^ { - 2 ( 3 ) } - e ^ { - ( 3 ) } + 2 ( c e ^ { - 2 ( 3 ) } + e ^ { - ( 3 ) } ) = e ^ { - ( 3 ) }$$

Solve for c (complex solution)

$$c\in \mathrm{C}$$

Solution Steps

Multiply -2 and 3 to get -6.

$$-2ce^{-6}-e^{-3}+2\left(ce^{-2\times 3}+e^{-3}\right)=e^{-3}$$

Multiply -2 and 3 to get -6.

$$-2ce^{-6}-e^{-3}+2\left(ce^{-6}+e^{-3}\right)=e^{-3}$$

Use the distributive property to multiply 2 by ce^{-6}+e^{-3}.

$$-2ce^{-6}-e^{-3}+2ce^{-6}+2e^{-3}=e^{-3}$$

Subtract 2e^{-3} from both sides.

$$-2ce^{-6}-e^{-3}+2ce^{-6}=e^{-3}-2e^{-3}$$

Combine e^{-3} and -2e^{-3} to get -e^{-3}.

$$-2ce^{-6}-e^{-3}+2ce^{-6}=-e^{-3}$$

Combine -2ce^{-6} and 2ce^{-6} to get 0.

$$-e^{-3}=-e^{-3}$$

Cancel out -1 on both sides.

$$e^{-3}=e^{-3}$$

This is true for any c.

$$c\in \mathrm{C}$$

Solve for c

$$c\in \mathrm{R}$$

Solution Steps

Multiply -2 and 3 to get -6.

$$-2ce^{-6}-e^{-3}+2\left(ce^{-2\times 3}+e^{-3}\right)=e^{-3}$$

Multiply -2 and 3 to get -6.

$$-2ce^{-6}-e^{-3}+2\left(ce^{-6}+e^{-3}\right)=e^{-3}$$

Use the distributive property to multiply 2 by ce^{-6}+e^{-3}.

$$-2ce^{-6}-e^{-3}+2ce^{-6}+2e^{-3}=e^{-3}$$

Subtract 2e^{-3} from both sides.

$$-2ce^{-6}-e^{-3}+2ce^{-6}=e^{-3}-2e^{-3}$$

Combine e^{-3} and -2e^{-3} to get -e^{-3}.

$$-2ce^{-6}-e^{-3}+2ce^{-6}=-e^{-3}$$

Combine -2ce^{-6} and 2ce^{-6} to get 0.

$$-e^{-3}=-e^{-3}$$

Cancel out -1 on both sides.

$$e^{-3}=e^{-3}$$

This is true for any c.

$$c\in \mathrm{R}$$