Solve (2-x)+2(x-2)(2-x)=(1-x)(x-2)

$$( 2 - x ) + 2 ( x - 2 ) ( 2 - x ) = ( 1 - x ) ( x - 2 )$$

Solve for x

$$x=2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by x-2.

$$2-x+\left(2x-4\right)\left(2-x\right)=\left(1-x\right)\left(x-2\right)$$

Use the distributive property to multiply 2x-4 by 2-x and combine like terms.

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

Combine -x and 8x to get 7x.

$$2+7x-2x^{2}-8=\left(1-x\right)\left(x-2\right)$$

Subtract 8 from 2 to get -6.

$$-6+7x-2x^{2}=\left(1-x\right)\left(x-2\right)$$

Use the distributive property to multiply 1-x by x-2 and combine like terms.

$$-6+7x-2x^{2}=3x-2-x^{2}$$

Subtract 3x from both sides.

$$-6+7x-2x^{2}-3x=-2-x^{2}$$

Combine 7x and -3x to get 4x.

$$-6+4x-2x^{2}=-2-x^{2}$$

Subtract -2 from both sides.

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

The opposite of -2 is 2.

$$-6+4x-2x^{2}+2=-x^{2}$$

Add x^{2} to both sides.

$$-6+4x-2x^{2}+2+x^{2}=0$$

Add -6 and 2 to get -4.

$$-4+4x-2x^{2}+x^{2}=0$$

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

$$-4+4x-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+4x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\left(-4\right)}}{2\left(-1\right)}$$

Multiply 4 times -4.

$$x=\frac{-4±\sqrt{16-16}}{2\left(-1\right)}$$

Add 16 to -16.

$$x=\frac{-4±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$x=-\frac{4}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=-\frac{4}{-2}$$

Divide -4 by -2.

$$x=2$$

Steps for Completing the Square

Use the distributive property to multiply 2 by x-2.

$$2-x+\left(2x-4\right)\left(2-x\right)=\left(1-x\right)\left(x-2\right)$$

Use the distributive property to multiply 2x-4 by 2-x and combine like terms.

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

Combine -x and 8x to get 7x.

$$2+7x-2x^{2}-8=\left(1-x\right)\left(x-2\right)$$

Subtract 8 from 2 to get -6.

$$-6+7x-2x^{2}=\left(1-x\right)\left(x-2\right)$$

Use the distributive property to multiply 1-x by x-2 and combine like terms.

$$-6+7x-2x^{2}=3x-2-x^{2}$$

Subtract 3x from both sides.

$$-6+7x-2x^{2}-3x=-2-x^{2}$$

Combine 7x and -3x to get 4x.

$$-6+4x-2x^{2}=-2-x^{2}$$

Add x^{2} to both sides.

$$-6+4x-2x^{2}+x^{2}=-2$$

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

$$-6+4x-x^{2}=-2$$

Add 6 to both sides.

$$4x-x^{2}=-2+6$$

Add -2 and 6 to get 4.

$$4x-x^{2}=4$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=\frac{4}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=\frac{4}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=\frac{4}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=-4$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -2.

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

Add -4 to 4.

$$x^{2}-4x+4=0$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-2=0 x-2=0$$

Add 2 to both sides of the equation.

$$x=2 x=2$$

The equation is now solved. Solutions are the same.

$$x=2$$