Solve (x+b)(2x-1)=2x^2+ax-2

$$( x + b ) ( 2 x - 1 ) = 2 x ^ { 2 } + a x - 2$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{-2bx+x+b-2}{x}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&b=2\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+b by 2x-1.

$$2x^{2}-x+2bx-b=2x^{2}+ax-2$$

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

$$2x^{2}+ax-2=2x^{2}-x+2bx-b$$

Subtract 2x^{2} from both sides.

$$ax-2=2x^{2}-x+2bx-b-2x^{2}$$

Combine 2x^{2} and -2x^{2} to get 0.

$$ax-2=-x+2bx-b$$

Add 2 to both sides.

$$ax=-x+2bx-b+2$$

The equation is in standard form.

$$xa=2bx-x-b+2$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{2bx-x-b+2}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{2bx-x-b+2}{x}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{2-x-ax}{2x-1}\text{, }&x\neq \frac{1}{2}\\b\in \mathrm{C}\text{, }&x=\frac{1}{2}\text{ and }a=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+b by 2x-1.

$$2x^{2}-x+2bx-b=2x^{2}+ax-2$$

Subtract 2x^{2} from both sides.

$$-x+2bx-b=2x^{2}+ax-2-2x^{2}$$

Combine 2x^{2} and -2x^{2} to get 0.

$$-x+2bx-b=ax-2$$

Add x to both sides.

$$2bx-b=ax-2+x$$

Combine all terms containing b.

$$\left(2x-1\right)b=ax-2+x$$

The equation is in standard form.

$$\left(2x-1\right)b=ax+x-2$$

Divide both sides by 2x-1.

$$\frac{\left(2x-1\right)b}{2x-1}=\frac{ax+x-2}{2x-1}$$

Dividing by 2x-1 undoes the multiplication by 2x-1.

$$b=\frac{ax+x-2}{2x-1}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{-2bx+x+b-2}{x}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&b=2\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+b by 2x-1.

$$2x^{2}-x+2bx-b=2x^{2}+ax-2$$

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

$$2x^{2}+ax-2=2x^{2}-x+2bx-b$$

Subtract 2x^{2} from both sides.

$$ax-2=2x^{2}-x+2bx-b-2x^{2}$$

Combine 2x^{2} and -2x^{2} to get 0.

$$ax-2=-x+2bx-b$$

Add 2 to both sides.

$$ax=-x+2bx-b+2$$

The equation is in standard form.

$$xa=2bx-x-b+2$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{2bx-x-b+2}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{2bx-x-b+2}{x}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{2-x-ax}{2x-1}\text{, }&x\neq \frac{1}{2}\\b\in \mathrm{R}\text{, }&x=\frac{1}{2}\text{ and }a=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+b by 2x-1.

$$2x^{2}-x+2bx-b=2x^{2}+ax-2$$

Subtract 2x^{2} from both sides.

$$-x+2bx-b=2x^{2}+ax-2-2x^{2}$$

Combine 2x^{2} and -2x^{2} to get 0.

$$-x+2bx-b=ax-2$$

Add x to both sides.

$$2bx-b=ax-2+x$$

Combine all terms containing b.

$$\left(2x-1\right)b=ax-2+x$$

The equation is in standard form.

$$\left(2x-1\right)b=ax+x-2$$

Divide both sides by 2x-1.

$$\frac{\left(2x-1\right)b}{2x-1}=\frac{ax+x-2}{2x-1}$$

Dividing by 2x-1 undoes the multiplication by 2x-1.

$$b=\frac{ax+x-2}{2x-1}$$