Solve isqrt{2}g(x-1)=2x^2-3x-1

$$i \sqrt { 2 } g ( x - 1 ) = 2 x ^ { 2 } - 3 x - 1$$

Solve for g

$$g=-\frac{\sqrt{2}\left(1+3x-2x^{2}\right)}{2\left(ix-i\right)}$$

$$x\neq 1$$

Steps for Solving Linear Equation

Use the distributive property to multiply i\sqrt{2}g by x-1.

$$i\sqrt{2}gx-ig\sqrt{2}=2x^{2}-3x-1$$

Reorder the terms.

$$\sqrt{2}igx-\sqrt{2}ig=2x^{2}-3x-1$$

Multiply -1 and i to get -i.

$$\sqrt{2}igx-i\sqrt{2}g=2x^{2}-3x-1$$

Combine all terms containing g.

$$\left(\sqrt{2}ix-i\sqrt{2}\right)g=2x^{2}-3x-1$$

Reorder the terms.

$$g\left(\sqrt{2}ix-\sqrt{2}i\right)=2x^{2}-3x-1$$

The equation is in standard form.

$$\left(\sqrt{2}ix-\sqrt{2}i\right)g=2x^{2}-3x-1$$

Divide both sides by i\sqrt{2}x-i\sqrt{2}.

$$\frac{\left(\sqrt{2}ix-\sqrt{2}i\right)g}{\sqrt{2}ix-\sqrt{2}i}=\frac{2x^{2}-3x-1}{\sqrt{2}ix-\sqrt{2}i}$$

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

$$g=\frac{2x^{2}-3x-1}{\sqrt{2}ix-\sqrt{2}i}$$

Divide 2x^{2}-3x-1 by i\sqrt{2}x-i\sqrt{2}.

$$g=\frac{\sqrt{2}\left(2x^{2}-3x-1\right)}{2\left(ix-i\right)}$$

Solve for x

$$x=\frac{-\sqrt{-2g^{2}-2\sqrt{2}ig+17}+\sqrt{2}ig+3}{4}$$

$$x=\frac{\sqrt{-2g^{2}-2\sqrt{2}ig+17}+\sqrt{2}ig+3}{4}$$