Solve f(x)=3x^2-5x-2quadyquadg(x)=2x-3

$$f ( x ) = 3 x ^ { 2 } - 5 x - 2 \quad y \quad g ( x ) = 2 x - 3$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=\frac{3x^{2}-7x+3}{2xy}\text{, }&x\neq 0\text{ and }y\neq 0\\g\in \mathrm{C}\text{, }&\left(x=\frac{7-\sqrt{13}}{6}\text{ or }x=\frac{\sqrt{13}+7}{6}\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 3x^{2} from both sides.

$$-5x-2ygx=2x-3-3x^{2}$$

Add 5x to both sides.

$$-2ygx=2x-3-3x^{2}+5x$$

Combine 2x and 5x to get 7x.

$$-2ygx=7x-3-3x^{2}$$

The equation is in standard form.

$$\left(-2xy\right)g=-3x^{2}+7x-3$$

Divide both sides by -2yx.

$$\frac{\left(-2xy\right)g}{-2xy}=\frac{-3x^{2}+7x-3}{-2xy}$$

Dividing by -2yx undoes the multiplication by -2yx.

$$g=\frac{-3x^{2}+7x-3}{-2xy}$$

Divide -3x^{2}+7x-3 by -2yx.

$$g=-\frac{-3x^{2}+7x-3}{2xy}$$

Solve for g

$$\left\{\begin{matrix}g=\frac{3x^{2}-7x+3}{2xy}\text{, }&x\neq 0\text{ and }y\neq 0\\g\in \mathrm{R}\text{, }&\left(x=\frac{7-\sqrt{13}}{6}\text{ or }x=\frac{\sqrt{13}+7}{6}\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 3x^{2} from both sides.

$$-5x-2ygx=2x-3-3x^{2}$$

Add 5x to both sides.

$$-2ygx=2x-3-3x^{2}+5x$$

Combine 2x and 5x to get 7x.

$$-2ygx=7x-3-3x^{2}$$

The equation is in standard form.

$$\left(-2xy\right)g=-3x^{2}+7x-3$$

Divide both sides by -2yx.

$$\frac{\left(-2xy\right)g}{-2xy}=\frac{-3x^{2}+7x-3}{-2xy}$$

Dividing by -2yx undoes the multiplication by -2yx.

$$g=\frac{-3x^{2}+7x-3}{-2xy}$$

Divide 7x-3-3x^{2} by -2yx.

$$g=-\frac{-3x^{2}+7x-3}{2xy}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{28gy+4\left(gy\right)^{2}+13}}{6}+\frac{gy}{3}+\frac{7}{6}$$

$$x=-\frac{\sqrt{28gy+4\left(gy\right)^{2}+13}}{6}+\frac{gy}{3}+\frac{7}{6}$$

Solve for x

$$x=\frac{\sqrt{28gy+4\left(gy\right)^{2}+13}}{6}+\frac{gy}{3}+\frac{7}{6}$$

$$x=-\frac{\sqrt{28gy+4\left(gy\right)^{2}+13}}{6}+\frac{gy}{3}+\frac{7}{6}\text{, }g=0\text{ or }y\geq \frac{3|g|}{g^{2}}-\frac{7}{2g}\text{ or }y\leq -\frac{3|g|}{g^{2}}-\frac{7}{2g}$$