Solve -13-x=(-4/3(x-1)^2+16/3)x7

$$- 13 - x = ( - \frac { 4 } { 3 } ( x - 1 ) ^ { 2 } + \frac { 16 } { 3 } ) x 7$$

Solve for x_7 (complex solution)

$$x_{7}=-\frac{3\left(x+13\right)}{4\left(3-x\right)\left(x+1\right)}$$

$$x\neq -1\text{ and }x\neq 3$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$-13-x=\left(-\frac{4}{3}\left(x^{2}-2x+1\right)+\frac{16}{3}\right)x_{7}$$

Use the distributive property to multiply -\frac{4}{3} by x^{2}-2x+1.

$$-13-x=\left(-\frac{4}{3}x^{2}+\frac{8}{3}x-\frac{4}{3}+\frac{16}{3}\right)x_{7}$$

Add -\frac{4}{3} and \frac{16}{3} to get 4.

$$-13-x=\left(-\frac{4}{3}x^{2}+\frac{8}{3}x+4\right)x_{7}$$

Use the distributive property to multiply -\frac{4}{3}x^{2}+\frac{8}{3}x+4 by x_{7}.

$$-13-x=-\frac{4}{3}x^{2}x_{7}+\frac{8}{3}xx_{7}+4x_{7}$$

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

$$-\frac{4}{3}x^{2}x_{7}+\frac{8}{3}xx_{7}+4x_{7}=-13-x$$

Combine all terms containing x_{7}.

$$\left(-\frac{4}{3}x^{2}+\frac{8}{3}x+4\right)x_{7}=-13-x$$

The equation is in standard form.

$$\left(-\frac{4x^{2}}{3}+\frac{8x}{3}+4\right)x_{7}=-x-13$$

Divide both sides by -\frac{4}{3}x^{2}+\frac{8}{3}x+4.

$$\frac{\left(-\frac{4x^{2}}{3}+\frac{8x}{3}+4\right)x_{7}}{-\frac{4x^{2}}{3}+\frac{8x}{3}+4}=\frac{-x-13}{-\frac{4x^{2}}{3}+\frac{8x}{3}+4}$$

Dividing by -\frac{4}{3}x^{2}+\frac{8}{3}x+4 undoes the multiplication by -\frac{4}{3}x^{2}+\frac{8}{3}x+4.

$$x_{7}=\frac{-x-13}{-\frac{4x^{2}}{3}+\frac{8x}{3}+4}$$

Divide -13-x by -\frac{4}{3}x^{2}+\frac{8}{3}x+4.

$$x_{7}=\frac{3\left(x+13\right)}{4\left(x-3\right)\left(x+1\right)}$$

Solve for x_7

$$x_{7}=-\frac{3\left(x+13\right)}{4\left(3-x\right)\left(x+1\right)}$$

$$x\neq 3\text{ and }x\neq -1$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$-13-x=\left(-\frac{4}{3}\left(x^{2}-2x+1\right)+\frac{16}{3}\right)x_{7}$$

Use the distributive property to multiply -\frac{4}{3} by x^{2}-2x+1.

$$-13-x=\left(-\frac{4}{3}x^{2}+\frac{8}{3}x-\frac{4}{3}+\frac{16}{3}\right)x_{7}$$

Add -\frac{4}{3} and \frac{16}{3} to get 4.

$$-13-x=\left(-\frac{4}{3}x^{2}+\frac{8}{3}x+4\right)x_{7}$$

Use the distributive property to multiply -\frac{4}{3}x^{2}+\frac{8}{3}x+4 by x_{7}.

$$-13-x=-\frac{4}{3}x^{2}x_{7}+\frac{8}{3}xx_{7}+4x_{7}$$

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

$$-\frac{4}{3}x^{2}x_{7}+\frac{8}{3}xx_{7}+4x_{7}=-13-x$$

Combine all terms containing x_{7}.

$$\left(-\frac{4}{3}x^{2}+\frac{8}{3}x+4\right)x_{7}=-13-x$$

The equation is in standard form.

$$\left(-\frac{4x^{2}}{3}+\frac{8x}{3}+4\right)x_{7}=-x-13$$

Divide both sides by -\frac{4}{3}x^{2}+\frac{8}{3}x+4.

$$\frac{\left(-\frac{4x^{2}}{3}+\frac{8x}{3}+4\right)x_{7}}{-\frac{4x^{2}}{3}+\frac{8x}{3}+4}=\frac{-x-13}{-\frac{4x^{2}}{3}+\frac{8x}{3}+4}$$

Dividing by -\frac{4}{3}x^{2}+\frac{8}{3}x+4 undoes the multiplication by -\frac{4}{3}x^{2}+\frac{8}{3}x+4.

$$x_{7}=\frac{-x-13}{-\frac{4x^{2}}{3}+\frac{8x}{3}+4}$$

Divide -13-x by -\frac{4}{3}x^{2}+\frac{8}{3}x+4.

$$x_{7}=\frac{3\left(x+13\right)}{4\left(x-3\right)\left(x+1\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{256x_{7}^{2}+672x_{7}+9}+8x_{7}+3}{8x_{7}}\text{; }x=\frac{-\sqrt{256x_{7}^{2}+672x_{7}+9}+8x_{7}+3}{8x_{7}}\text{, }&x_{7}\neq 0\\x=-13\text{, }&x_{7}=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{256x_{7}^{2}+672x_{7}+9}+8x_{7}+3}{8x_{7}}\text{; }x=\frac{-\sqrt{256x_{7}^{2}+672x_{7}+9}+8x_{7}+3}{8x_{7}}\text{, }&x_{7}\leq -\frac{3\sqrt{3}}{4}-\frac{21}{16}\text{ or }\left(x_{7}\neq 0\text{ and }x_{7}\geq \frac{3\sqrt{3}}{4}-\frac{21}{16}\right)\\x=-13\text{, }&x_{7}=0\end{matrix}\right.$$