Solve (2-x)^2+(1-x)^2=(3-x)^2

$$( 2 - x ) ^ { 2 } + ( 1 - x ) ^ { 2 } = ( 3 - x ) ^ { 2 }$$

Solve for x

$$x=2$$

$$x=-2$$

Steps Using Difference of Squares

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-x\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-x\right)^{2}.

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

Add 4 and 1 to get 5.

$$5-4x+x^{2}-2x+x^{2}=\left(3-x\right)^{2}$$

Combine -4x and -2x to get -6x.

$$5-6x+x^{2}+x^{2}=\left(3-x\right)^{2}$$

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

$$5-6x+2x^{2}=\left(3-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-x\right)^{2}.

$$5-6x+2x^{2}=9-6x+x^{2}$$

Subtract 9 from both sides.

$$5-6x+2x^{2}-9=-6x+x^{2}$$

Subtract 9 from 5 to get -4.

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

Add 6x to both sides.

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

Combine -6x and 6x to get 0.

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

Subtract x^{2} from both sides.

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

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

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

Consider -4+x^{2}. Rewrite -4+x^{2} as x^{2}-2^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-2\right)\left(x+2\right)=0$$

To find equation solutions, solve x-2=0 and x+2=0.

$$x=2 x=-2$$

Steps by Finding Square Root

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-x\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-x\right)^{2}.

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

Add 4 and 1 to get 5.

$$5-4x+x^{2}-2x+x^{2}=\left(3-x\right)^{2}$$

Combine -4x and -2x to get -6x.

$$5-6x+x^{2}+x^{2}=\left(3-x\right)^{2}$$

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

$$5-6x+2x^{2}=\left(3-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-x\right)^{2}.

$$5-6x+2x^{2}=9-6x+x^{2}$$

Add 6x to both sides.

$$5-6x+2x^{2}+6x=9+x^{2}$$

Combine -6x and 6x to get 0.

$$5+2x^{2}=9+x^{2}$$

Subtract x^{2} from both sides.

$$5+2x^{2}-x^{2}=9$$

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

$$5+x^{2}=9$$

Subtract 5 from both sides.

$$x^{2}=9-5$$

Subtract 5 from 9 to get 4.

$$x^{2}=4$$

Take the square root of both sides of the equation.

$$x=2 x=-2$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-x\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-x\right)^{2}.

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

Add 4 and 1 to get 5.

$$5-4x+x^{2}-2x+x^{2}=\left(3-x\right)^{2}$$

Combine -4x and -2x to get -6x.

$$5-6x+x^{2}+x^{2}=\left(3-x\right)^{2}$$

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

$$5-6x+2x^{2}=\left(3-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-x\right)^{2}.

$$5-6x+2x^{2}=9-6x+x^{2}$$

Subtract 9 from both sides.

$$5-6x+2x^{2}-9=-6x+x^{2}$$

Subtract 9 from 5 to get -4.

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

Add 6x to both sides.

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

Combine -6x and 6x to get 0.

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

Subtract x^{2} from both sides.

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

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

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

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

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

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

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

Square 0.

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

Multiply -4 times -4.

$$x=\frac{0±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{0±4}{2}$$

Now solve the equation x=\frac{0±4}{2} when ± is plus. Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{0±4}{2} when ± is minus. Divide -4 by 2.

$$x=-2$$

The equation is now solved.

$$x=2 x=-2$$