Solve Deltaf=-3*(x_{0}+4x)^2+4(x_{0}-0x)-(-3x_0^2+4x_{0})=

$$\Delta f = - 3 \cdot ( x _ { 0 } + 4 x ) ^ { 2 } + 4 ( x _ { 0 } - 0 x ) - ( - 3 x _ { 0 } ^ { 2 } + 4 x _ { 0 } ) =$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=-\frac{24x\left(2x+x_{0}\right)}{\Delta }\text{, }&\Delta \neq 0\\f\in \mathrm{C}\text{, }&\left(x_{0}=-2x\text{ or }x=0\right)\text{ and }\Delta =0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x_{0}+4x\right)^{2}.

$$\Delta f=-3\left(x_{0}^{2}+8x_{0}x+16x^{2}\right)+4\left(x_{0}-0x\right)-\left(-3x_{0}^{2}+4x_{0}\right)$$

Use the distributive property to multiply -3 by x_{0}^{2}+8x_{0}x+16x^{2}.

$$\Delta f=-3x_{0}^{2}-24xx_{0}-48x^{2}+4\left(x_{0}-0x\right)-\left(-3x_{0}^{2}+4x_{0}\right)$$

Anything times zero gives zero.

$$\Delta f=-3x_{0}^{2}-24xx_{0}-48x^{2}+4\left(x_{0}-0\right)-\left(-3x_{0}^{2}+4x_{0}\right)$$

To find the opposite of -3x_{0}^{2}+4x_{0}, find the opposite of each term.

$$\Delta f=-3x_{0}^{2}-24xx_{0}-48x^{2}+4\left(x_{0}-0\right)+3x_{0}^{2}-4x_{0}$$

Combine -3x_{0}^{2} and 3x_{0}^{2} to get 0.

$$\Delta f=-24xx_{0}-48x^{2}+4\left(x_{0}-0\right)-4x_{0}$$

Reorder the terms.

$$f\Delta =-48x^{2}-24xx_{0}-4x_{0}+4x_{0}$$

Combine -4x_{0} and 4x_{0} to get 0.

$$f\Delta =-48x^{2}-24xx_{0}$$

The equation is in standard form.

$$\Delta f=-48x^{2}-24xx_{0}$$

Divide both sides by \Delta .

$$\frac{\Delta f}{\Delta }=-\frac{24x\left(2x+x_{0}\right)}{\Delta }$$

Dividing by \Delta undoes the multiplication by \Delta .

$$f=-\frac{24x\left(2x+x_{0}\right)}{\Delta }$$

Solve for f

$$\left\{\begin{matrix}f=-\frac{24x\left(2x+x_{0}\right)}{\Delta }\text{, }&\Delta \neq 0\\f\in \mathrm{R}\text{, }&\left(x_{0}=-2x\text{ or }x=0\right)\text{ and }\Delta =0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x_{0}+4x\right)^{2}.

$$\Delta f=-3\left(x_{0}^{2}+8x_{0}x+16x^{2}\right)+4\left(x_{0}-0x\right)-\left(-3x_{0}^{2}+4x_{0}\right)$$

Use the distributive property to multiply -3 by x_{0}^{2}+8x_{0}x+16x^{2}.

$$\Delta f=-3x_{0}^{2}-24xx_{0}-48x^{2}+4\left(x_{0}-0x\right)-\left(-3x_{0}^{2}+4x_{0}\right)$$

Anything times zero gives zero.

$$\Delta f=-3x_{0}^{2}-24xx_{0}-48x^{2}+4\left(x_{0}-0\right)-\left(-3x_{0}^{2}+4x_{0}\right)$$

To find the opposite of -3x_{0}^{2}+4x_{0}, find the opposite of each term.

$$\Delta f=-3x_{0}^{2}-24xx_{0}-48x^{2}+4\left(x_{0}-0\right)+3x_{0}^{2}-4x_{0}$$

Combine -3x_{0}^{2} and 3x_{0}^{2} to get 0.

$$\Delta f=-24xx_{0}-48x^{2}+4\left(x_{0}-0\right)-4x_{0}$$

Reorder the terms.

$$f\Delta =-48x^{2}-24xx_{0}-4x_{0}+4x_{0}$$

Combine -4x_{0} and 4x_{0} to get 0.

$$f\Delta =-48x^{2}-24xx_{0}$$

The equation is in standard form.

$$\Delta f=-48x^{2}-24xx_{0}$$

Divide both sides by \Delta .

$$\frac{\Delta f}{\Delta }=-\frac{24x\left(2x+x_{0}\right)}{\Delta }$$

Dividing by \Delta undoes the multiplication by \Delta .

$$f=-\frac{24x\left(2x+x_{0}\right)}{\Delta }$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{9x_{0}^{2}-3f\Delta }}{12}-\frac{x_{0}}{4}$$

$$x=\frac{\sqrt{9x_{0}^{2}-3f\Delta }}{12}-\frac{x_{0}}{4}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{3f\Delta }}{12}\text{, }&\left(x_{0}=0\text{ and }\Delta =0\text{ and }f\neq 0\right)\text{ or }\left(\Delta =\frac{3x_{0}^{2}}{f}\text{ and }x_{0}\leq 0\text{ and }f\neq 0\right)\\x=-\frac{\sqrt{3f\Delta }}{12}\text{, }&\left(x_{0}=0\text{ and }\Delta =0\text{ and }f\neq 0\right)\text{ or }\left(\Delta =\frac{3x_{0}^{2}}{f}\text{ and }x_{0}\geq 0\text{ and }f\neq 0\right)\\x=-\frac{\sqrt{9x_{0}^{2}-3f\Delta }}{12}-\frac{x_{0}}{4}\text{; }x=\frac{\sqrt{9x_{0}^{2}-3f\Delta }}{12}-\frac{x_{0}}{4}\text{, }&\left(f>0\text{ and }\Delta \leq \frac{3x_{0}^{2}}{f}\right)\text{ or }\left(\Delta \geq \frac{3x_{0}^{2}}{f}\text{ and }f<0\right)\\x=\frac{-|x_{0}|-x_{0}}{4}\text{; }x=\frac{|x_{0}|-x_{0}}{4}\text{, }&f=0\end{matrix}\right.$$