Solve d(x-3)^2=12+x-5x^2

$$d ( x - 3 ) ^ { 2 } = 12 + x - 5 x ^ { 2 }$$

Solve for d

$$d=-\frac{5x^{2}-x-12}{\left(x-3\right)^{2}}$$

$$x\neq 3$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$d\left(x^{2}-6x+9\right)=12+x-5x^{2}$$

Use the distributive property to multiply d by x^{2}-6x+9.

$$dx^{2}-6dx+9d=12+x-5x^{2}$$

Combine all terms containing d.

$$\left(x^{2}-6x+9\right)d=12+x-5x^{2}$$

Divide both sides by x^{2}-6x+9.

$$\frac{\left(x^{2}-6x+9\right)d}{x^{2}-6x+9}=\frac{12+x-5x^{2}}{x^{2}-6x+9}$$

Dividing by x^{2}-6x+9 undoes the multiplication by x^{2}-6x+9.

$$d=\frac{12+x-5x^{2}}{x^{2}-6x+9}$$

Divide 12+x-5x^{2} by x^{2}-6x+9.

$$d=\frac{12+x-5x^{2}}{\left(x-3\right)^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{6d+\sqrt{241-120d}+1}{2\left(d+5\right)}\text{; }x=\frac{6d-\sqrt{241-120d}+1}{2\left(d+5\right)}\text{, }&d\neq -5\\x=\frac{57}{29}\text{, }&d=-5\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{6d+\sqrt{241-120d}+1}{2\left(d+5\right)}\text{; }x=\frac{6d-\sqrt{241-120d}+1}{2\left(d+5\right)}\text{, }&d\neq -5\text{ and }d\leq \frac{241}{120}\\x=\frac{57}{29}\text{, }&d=-5\end{matrix}\right.$$