Solve sqrt{2}(x+sqrt{2})=3x-2/sqrt{2}

$$\sqrt { 2 } ( x + \sqrt { 2 } ) = \frac { 3 x - 2 } { \sqrt { 2 } }$$

Solve for x

$$x=2\sqrt{2}+2\approx 4.828427125$$

Steps for Solving Linear Equation

Use the distributive property to multiply \sqrt{2} by x+\sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Rationalize the denominator of \frac{3x-2}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Use the distributive property to multiply 3x-2 by \sqrt{2}.

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

Subtract \frac{3x\sqrt{2}-2\sqrt{2}}{2} from both sides.

$$\sqrt{2}x+2-\frac{3x\sqrt{2}-2\sqrt{2}}{2}=0$$

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

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

Multiply both sides of the equation by 2.

$$2\sqrt{2}x-\left(3x\sqrt{2}-2\sqrt{2}\right)=-4$$

To find the opposite of 3x\sqrt{2}-2\sqrt{2}, find the opposite of each term.

$$2\sqrt{2}x-3x\sqrt{2}-\left(-2\sqrt{2}\right)=-4$$

The opposite of -2\sqrt{2} is 2\sqrt{2}.

$$2\sqrt{2}x-3x\sqrt{2}+2\sqrt{2}=-4$$

Combine 2\sqrt{2}x and -3x\sqrt{2} to get -\sqrt{2}x.

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

Subtract 2\sqrt{2} from both sides.

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

The equation is in standard form.

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

Divide both sides by -\sqrt{2}.

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

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

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

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

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