Solve (1-x)^2=1-0.5

$$( 1 - x ) ^ { 2 } = 1 - 0.5$$

Solve for x

$$x=-\frac{\sqrt{2}}{2}+1\approx 0.292893219$$

$$x=\frac{\sqrt{2}}{2}+1\approx 1.707106781$$

Steps Using the Quadratic Formula

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

$$1-2x+x^{2}=1-0.5$$

Subtract 0.5 from 1 to get 0.5.

$$1-2x+x^{2}=0.5$$

Subtract 0.5 from both sides.

$$1-2x+x^{2}-0.5=0$$

Subtract 0.5 from 1 to get 0.5.

$$0.5-2x+x^{2}=0$$

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.

$$x^{2}-2x+\frac{1}{2}=0$$

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

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

Square -2.

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

Multiply -4 times \frac{1}{2}.

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

Add 4 to -2.

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

The opposite of -2 is 2.

$$x=\frac{2±\sqrt{2}}{2}$$

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

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

Divide 2+\sqrt{2} by 2.

$$x=\frac{\sqrt{2}}{2}+1$$

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

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

Divide 2-\sqrt{2} by 2.

$$x=-\frac{\sqrt{2}}{2}+1$$

The equation is now solved.

$$x=\frac{\sqrt{2}}{2}+1 x=-\frac{\sqrt{2}}{2}+1$$

Steps for Completing the Square

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

$$1-2x+x^{2}=1-0.5$$

Subtract 0.5 from 1 to get 0.5.

$$1-2x+x^{2}=0.5$$

Subtract 1 from both sides.

$$-2x+x^{2}=0.5-1$$

Subtract 1 from 0.5 to get -0.5.

$$-2x+x^{2}=-0.5$$

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.

$$x^{2}-2x=-0.5$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-0.5+1$$

Add -0.5 to 1.

$$x^{2}-2x+1=0.5$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=0.5$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{0.5}$$

Simplify.

$$x-1=\frac{\sqrt{2}}{2} x-1=-\frac{\sqrt{2}}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{2}}{2}+1 x=-\frac{\sqrt{2}}{2}+1$$