Solve 4+sqrt{2}x^2=2

$$4+ \sqrt{ 2 } { x }^{ 2 } =2$$

Solve for x (complex solution)

$$x=-\sqrt[4]{2}i\approx -0-1.189207115i$$

$$x=\sqrt[4]{2}i\approx 1.189207115i$$

Steps by Finding Square Root

Subtract 4 from both sides.

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

Subtract 4 from 2 to get -2.

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

Dividing by \sqrt{2} undoes the multiplication by \sqrt{2}.

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

Divide -2 by \sqrt{2}.

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

Take the square root of both sides of the equation.

$$x=\sqrt[4]{2}i x=-\sqrt[4]{2}i$$

Steps Using the Quadratic Formula

Subtract 2 from both sides.

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

Subtract 2 from 4 to get 2.

$$2+\sqrt{2}x^{2}=0$$

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.

$$\sqrt{2}x^{2}+2=0$$

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

$$x=\frac{0±\sqrt{0^{2}-4\sqrt{2}\times 2}}{2\sqrt{2}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\sqrt{2}\times 2}}{2\sqrt{2}}$$

Multiply -4 times \sqrt{2}.

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

Multiply -4\sqrt{2} times 2.

$$x=\frac{0±\sqrt{-8\sqrt{2}}}{2\sqrt{2}}$$

Take the square root of -8\sqrt{2}.

$$x=\frac{0±2\times 2^{\frac{3}{4}}i}{2\sqrt{2}}$$

Now solve the equation x=\frac{0±2\times 2^{\frac{3}{4}}i}{2\sqrt{2}} when ± is plus.

$$x=\frac{2i}{2^{\frac{3}{4}}}$$

Now solve the equation x=\frac{0±2\times 2^{\frac{3}{4}}i}{2\sqrt{2}} when ± is minus.

$$x=-\sqrt[4]{2}i$$

The equation is now solved.

$$x=\frac{2i}{2^{\frac{3}{4}}} x=-\sqrt[4]{2}i$$