Solve x^2+(sqrt{2})^2=(2x)^2

$$x ^ { 2 } + ( \sqrt { 2 } ) ^ { 2 } = ( 2 x ) ^ { 2 }$$

Solve for x

$$x=\frac{\sqrt{6}}{3}\approx 0.816496581$$

$$x=-\frac{\sqrt{6}}{3}\approx -0.816496581$$

Steps by Finding Square Root

The square of \sqrt{2} is 2.

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

Expand \left(2x\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Subtract 4x^{2} from both sides.

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

Combine x^{2} and -4x^{2} to get -3x^{2}.

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

Subtract 2 from both sides. Anything subtracted from zero gives its negation.

$$-3x^{2}=-2$$

Divide both sides by -3.

$$x^{2}=\frac{-2}{-3}$$

Fraction \frac{-2}{-3} can be simplified to \frac{2}{3} by removing the negative sign from both the numerator and the denominator.

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

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{6}}{3} x=-\frac{\sqrt{6}}{3}$$

Steps Using the Quadratic Formula

The square of \sqrt{2} is 2.

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

Expand \left(2x\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Subtract 4x^{2} from both sides.

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

Combine x^{2} and -4x^{2} to get -3x^{2}.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 0 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-3\right)\times 2}}{2\left(-3\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-3\right)\times 2}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{0±\sqrt{12\times 2}}{2\left(-3\right)}$$

Multiply 12 times 2.

$$x=\frac{0±\sqrt{24}}{2\left(-3\right)}$$

Take the square root of 24.

$$x=\frac{0±2\sqrt{6}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{0±2\sqrt{6}}{-6}$$

Now solve the equation x=\frac{0±2\sqrt{6}}{-6} when ± is plus.

$$x=-\frac{\sqrt{6}}{3}$$

Now solve the equation x=\frac{0±2\sqrt{6}}{-6} when ± is minus.

$$x=\frac{\sqrt{6}}{3}$$

The equation is now solved.

$$x=-\frac{\sqrt{6}}{3} x=\frac{\sqrt{6}}{3}$$