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

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

Solve for x

$$x=\sqrt{2}+3\approx 4.414213562$$

$$x=3-\sqrt{2}\approx 1.585786438$$

Steps Using the Quadratic Formula

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

$$9-6x+x^{2}+\left(2-x\right)^{2}+\left(-1+x\right)^{2}=x^{2}$$

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

$$9-6x+x^{2}+4-4x+x^{2}+\left(-1+x\right)^{2}=x^{2}$$

Add 9 and 4 to get 13.

$$13-6x+x^{2}-4x+x^{2}+\left(-1+x\right)^{2}=x^{2}$$

Combine -6x and -4x to get -10x.

$$13-10x+x^{2}+x^{2}+\left(-1+x\right)^{2}=x^{2}$$

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

$$13-10x+2x^{2}+\left(-1+x\right)^{2}=x^{2}$$

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

$$13-10x+2x^{2}+1-2x+x^{2}=x^{2}$$

Add 13 and 1 to get 14.

$$14-10x+2x^{2}-2x+x^{2}=x^{2}$$

Combine -10x and -2x to get -12x.

$$14-12x+2x^{2}+x^{2}=x^{2}$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$14-12x+3x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$14-12x+3x^{2}-x^{2}=0$$

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

$$14-12x+2x^{2}=0$$

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.

$$2x^{2}-12x+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -12 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 2\times 14}}{2\times 2}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 2\times 14}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-12\right)±\sqrt{144-8\times 14}}{2\times 2}$$

Multiply -8 times 14.

$$x=\frac{-\left(-12\right)±\sqrt{144-112}}{2\times 2}$$

Add 144 to -112.

$$x=\frac{-\left(-12\right)±\sqrt{32}}{2\times 2}$$

Take the square root of 32.

$$x=\frac{-\left(-12\right)±4\sqrt{2}}{2\times 2}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{12±4\sqrt{2}}{4}$$

Now solve the equation x=\frac{12±4\sqrt{2}}{4} when ± is plus. Add 12 to 4\sqrt{2}.

$$x=\frac{4\sqrt{2}+12}{4}$$

Divide 12+4\sqrt{2} by 4.

$$x=\sqrt{2}+3$$

Now solve the equation x=\frac{12±4\sqrt{2}}{4} when ± is minus. Subtract 4\sqrt{2} from 12.

$$x=\frac{12-4\sqrt{2}}{4}$$

Divide 12-4\sqrt{2} by 4.

$$x=3-\sqrt{2}$$

The equation is now solved.

$$x=\sqrt{2}+3 x=3-\sqrt{2}$$

Steps for Completing the Square

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

$$9-6x+x^{2}+\left(2-x\right)^{2}+\left(-1+x\right)^{2}=x^{2}$$

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

$$9-6x+x^{2}+4-4x+x^{2}+\left(-1+x\right)^{2}=x^{2}$$

Add 9 and 4 to get 13.

$$13-6x+x^{2}-4x+x^{2}+\left(-1+x\right)^{2}=x^{2}$$

Combine -6x and -4x to get -10x.

$$13-10x+x^{2}+x^{2}+\left(-1+x\right)^{2}=x^{2}$$

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

$$13-10x+2x^{2}+\left(-1+x\right)^{2}=x^{2}$$

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

$$13-10x+2x^{2}+1-2x+x^{2}=x^{2}$$

Add 13 and 1 to get 14.

$$14-10x+2x^{2}-2x+x^{2}=x^{2}$$

Combine -10x and -2x to get -12x.

$$14-12x+2x^{2}+x^{2}=x^{2}$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$14-12x+3x^{2}=x^{2}$$

Subtract x^{2} from both sides.

$$14-12x+3x^{2}-x^{2}=0$$

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

$$14-12x+2x^{2}=0$$

Subtract 14 from both sides. Anything subtracted from zero gives its negation.

$$-12x+2x^{2}=-14$$

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.

$$2x^{2}-12x=-14$$

Divide both sides by 2.

$$\frac{2x^{2}-12x}{2}=-\frac{14}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{12}{2}\right)x=-\frac{14}{2}$$

Divide -12 by 2.

$$x^{2}-6x=-\frac{14}{2}$$

Divide -14 by 2.

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

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-7+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-7+9$$

Add -7 to 9.

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

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{2}$$

Simplify.

$$x-3=\sqrt{2} x-3=-\sqrt{2}$$

Add 3 to both sides of the equation.

$$x=\sqrt{2}+3 x=3-\sqrt{2}$$