Solve Deltaf/Deltax=[(x+Deltax)^2-(x)(x+Deltax)+6]/Deltax

$$\frac { \Delta f } { \Delta x } = \frac { [ ( x + \Delta x ) ^ { 2 } - ( x ) ( x + \Delta x ) + 6 ] } { \Delta x }$$

Solve for f

$$f=\frac{\Delta x^{2}+\left(x\Delta \right)^{2}+6}{\Delta }$$

$$\Delta \neq 0\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\Delta .

$$\Delta f=\left(x+\Delta x\right)^{2}-x\left(x+\Delta x\right)+6$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+\Delta x\right)^{2}.

$$\Delta f=x^{2}+2x\Delta x+\Delta ^{2}x^{2}-x\left(x+\Delta x\right)+6$$

Multiply x and x to get x^{2}.

$$\Delta f=x^{2}+2x^{2}\Delta +\Delta ^{2}x^{2}-x\left(x+\Delta x\right)+6$$

Use the distributive property to multiply x by x+\Delta x.

$$\Delta f=x^{2}+2x^{2}\Delta +\Delta ^{2}x^{2}-\left(x^{2}+\Delta x^{2}\right)+6$$

To find the opposite of x^{2}+\Delta x^{2}, find the opposite of each term.

$$\Delta f=x^{2}+2x^{2}\Delta +\Delta ^{2}x^{2}-x^{2}-\Delta x^{2}+6$$

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

$$\Delta f=2x^{2}\Delta +\Delta ^{2}x^{2}-\Delta x^{2}+6$$

Combine 2x^{2}\Delta and -\Delta x^{2} to get x^{2}\Delta .

$$\Delta f=x^{2}\Delta +\Delta ^{2}x^{2}+6$$

The equation is in standard form.

$$\Delta f=x^{2}\Delta ^{2}+\Delta x^{2}+6$$

Divide both sides by \Delta .

$$\frac{\Delta f}{\Delta }=\frac{x^{2}\Delta ^{2}+\Delta x^{2}+6}{\Delta }$$

Dividing by \Delta undoes the multiplication by \Delta .

$$f=\frac{x^{2}\Delta ^{2}+\Delta x^{2}+6}{\Delta }$$

Divide x^{2}\Delta +\Delta ^{2}x^{2}+6 by \Delta .

$$f=\Delta x^{2}+x^{2}+\frac{6}{\Delta }$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\Delta ^{-\frac{1}{2}}\left(\Delta +1\right)^{-\frac{1}{2}}\sqrt{f\Delta -6}\text{; }x=\Delta ^{-\frac{1}{2}}\left(\Delta +1\right)^{-\frac{1}{2}}\sqrt{f\Delta -6}\text{, }&\left(f=0\text{ or }\Delta \neq \frac{6}{f}\right)\text{ and }\Delta \neq -1\text{ and }\Delta \neq 0\\x\neq 0\text{, }&f=-6\text{ and }\Delta =-1\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\sqrt{\frac{f\Delta -6}{\Delta \left(\Delta +1\right)}}\text{; }x=-\sqrt{\frac{f\Delta -6}{\Delta \left(\Delta +1\right)}}\text{, }&\left(f>0\text{ and }\Delta >\frac{6}{f}\text{ and }\Delta >0\right)\text{ or }\left(\Delta >-1\text{ and }\Delta >\frac{6}{f}\text{ and }f<0\text{ and }\Delta <0\right)\text{ or }\left(\Delta <0\text{ and }\Delta >-1\text{ and }f=-6\right)\text{ or }\left(f<0\text{ and }\Delta <\frac{6}{f}\text{ and }\Delta >0\right)\text{ or }\left(f>0\text{ and }\Delta >\frac{6}{f}\text{ and }\Delta <-1\right)\text{ or }\left(\Delta >-1\text{ and }\Delta <\frac{6}{f}\text{ and }f>0\text{ and }\Delta <0\right)\text{ or }\left(f=0\text{ and }\Delta <0\text{ and }\Delta >-1\right)\text{ or }\left(\Delta <\frac{6}{f}\text{ and }f<0\text{ and }\Delta <-1\right)\\x\neq 0\text{, }&f=-6\text{ and }\Delta =-1\end{matrix}\right.$$