Solve (1-lambda)(lambda^2-4x+2)+2(1-2lambda)+psi(lambda-1)=0

$$( 1 - \lambda ) ( \lambda ^ { 2 } - 4 x + 2 ) + 2 ( 1 - 2 \lambda ) + \psi ( \lambda - 1 ) = 0$$

Solve for x

$$x=-\frac{4-\psi -6\lambda +\lambda \psi +\lambda ^{2}-\lambda ^{3}}{4\left(\lambda -1\right)}$$

$$\lambda \neq 1$$

Steps for Solving Linear Equation

Use the distributive property to multiply 1-\lambda by \lambda ^{2}-4x+2.

$$\lambda ^{2}-4x+2-\lambda ^{3}+4x\lambda -2\lambda +2\left(1-2\lambda \right)+\psi \left(\lambda -1\right)=0$$

Use the distributive property to multiply 2 by 1-2\lambda .

$$\lambda ^{2}-4x+2-\lambda ^{3}+4x\lambda -2\lambda +2-4\lambda +\psi \left(\lambda -1\right)=0$$

Add 2 and 2 to get 4.

$$\lambda ^{2}-4x+4-\lambda ^{3}+4x\lambda -2\lambda -4\lambda +\psi \left(\lambda -1\right)=0$$

Combine -2\lambda and -4\lambda to get -6\lambda .

$$\lambda ^{2}-4x+4-\lambda ^{3}+4x\lambda -6\lambda +\psi \left(\lambda -1\right)=0$$

Use the distributive property to multiply \psi by \lambda -1.

$$\lambda ^{2}-4x+4-\lambda ^{3}+4x\lambda -6\lambda +\psi \lambda -\psi =0$$

Subtract \lambda ^{2} from both sides. Anything subtracted from zero gives its negation.

$$-4x+4-\lambda ^{3}+4x\lambda -6\lambda +\psi \lambda -\psi =-\lambda ^{2}$$

Subtract 4 from both sides.

$$-4x-\lambda ^{3}+4x\lambda -6\lambda +\psi \lambda -\psi =-\lambda ^{2}-4$$

Add \lambda ^{3} to both sides.

$$-4x+4x\lambda -6\lambda +\psi \lambda -\psi =-\lambda ^{2}-4+\lambda ^{3}$$

Add 6\lambda to both sides.

$$-4x+4x\lambda +\psi \lambda -\psi =-\lambda ^{2}-4+\lambda ^{3}+6\lambda$$

Subtract \psi \lambda from both sides.

$$-4x+4x\lambda -\psi =-\lambda ^{2}-4+\lambda ^{3}+6\lambda -\psi \lambda$$

Add \psi to both sides.

$$-4x+4x\lambda =-\lambda ^{2}-4+\lambda ^{3}+6\lambda -\psi \lambda +\psi$$

Combine all terms containing x.

$$\left(-4+4\lambda \right)x=-\lambda ^{2}-4+\lambda ^{3}+6\lambda -\psi \lambda +\psi$$

The equation is in standard form.

$$\left(4\lambda -4\right)x=\lambda ^{3}-\lambda ^{2}-\lambda \psi +6\lambda +\psi -4$$

Divide both sides by 4\lambda -4.

$$\frac{\left(4\lambda -4\right)x}{4\lambda -4}=\frac{\lambda ^{3}-\lambda ^{2}-\lambda \psi +6\lambda +\psi -4}{4\lambda -4}$$

Dividing by 4\lambda -4 undoes the multiplication by 4\lambda -4.

$$x=\frac{\lambda ^{3}-\lambda ^{2}-\lambda \psi +6\lambda +\psi -4}{4\lambda -4}$$

Divide -\lambda ^{2}-4+\lambda ^{3}+6\lambda -\psi \lambda +\psi by 4\lambda -4.

$$x=\frac{\lambda ^{3}-\lambda ^{2}-\lambda \psi +6\lambda +\psi -4}{4\left(\lambda -1\right)}$$