Solve 0=-2(x-2)^2-1

$$0 = - 2 ( x - 2 ) ^ { 2 } - 1$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{2}i}{2}+2\approx 2-0.707106781i$$

$$x=\frac{\sqrt{2}i}{2}+2\approx 2+0.707106781i$$

Steps Using the Quadratic Formula

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

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

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

$$0=-2x^{2}+8x-8-1$$

Subtract 1 from -8 to get -9.

$$0=-2x^{2}+8x-9$$

Swap sides so that all variable terms are on the left hand side.

$$-2x^{2}+8x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 8 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-2\right)\left(-9\right)}}{2\left(-2\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-2\right)\left(-9\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-8±\sqrt{64+8\left(-9\right)}}{2\left(-2\right)}$$

Multiply 8 times -9.

$$x=\frac{-8±\sqrt{64-72}}{2\left(-2\right)}$$

Add 64 to -72.

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

Take the square root of -8.

$$x=\frac{-8±2\sqrt{2}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-8±2\sqrt{2}i}{-4}$$

Now solve the equation x=\frac{-8±2\sqrt{2}i}{-4} when ± is plus. Add -8 to 2i\sqrt{2}.

$$x=\frac{-8+2\sqrt{2}i}{-4}$$

Divide -8+2i\sqrt{2} by -4.

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

Now solve the equation x=\frac{-8±2\sqrt{2}i}{-4} when ± is minus. Subtract 2i\sqrt{2} from -8.

$$x=\frac{-2\sqrt{2}i-8}{-4}$$

Divide -8-2i\sqrt{2} by -4.

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

The equation is now solved.

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

Steps for Completing the Square

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

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

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

$$0=-2x^{2}+8x-8-1$$

Subtract 1 from -8 to get -9.

$$0=-2x^{2}+8x-9$$

Swap sides so that all variable terms are on the left hand side.

$$-2x^{2}+8x-9=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$-2x^{2}+8x=9$$

Divide both sides by -2.

$$\frac{-2x^{2}+8x}{-2}=\frac{9}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{8}{-2}x=\frac{9}{-2}$$

Divide 8 by -2.

$$x^{2}-4x=\frac{9}{-2}$$

Divide 9 by -2.

$$x^{2}-4x=-\frac{9}{2}$$

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.

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

Square -2.

$$x^{2}-4x+4=-\frac{9}{2}+4$$

Add -\frac{9}{2} to 4.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 2 to both sides of the equation.

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