Solve 7dx^2=x^2+412-7x+x^2

$$7 d x ^ { 2 } = x ^ { 2 } + 412 - 7 x + x ^ { 2 }$$

Solve for d

$$d=\frac{2}{7}-\frac{1}{x}+\frac{412}{7x^{2}}$$

$$x\neq 0$$

Steps for Solving Linear Equation

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

$$7dx^{2}=2x^{2}+412-7x$$

The equation is in standard form.

$$7x^{2}d=2x^{2}-7x+412$$

Divide both sides by 7x^{2}.

$$\frac{7x^{2}d}{7x^{2}}=\frac{2x^{2}-7x+412}{7x^{2}}$$

Dividing by 7x^{2} undoes the multiplication by 7x^{2}.

$$d=\frac{2x^{2}-7x+412}{7x^{2}}$$

Divide 2x^{2}+412-7x by 7x^{2}.

$$d=\frac{2}{7}+\frac{\frac{412}{7}-x}{x^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{11536d-3247}-7}{2\left(7d-2\right)}\text{; }x=-\frac{\sqrt{11536d-3247}+7}{2\left(7d-2\right)}\text{, }&d\neq \frac{2}{7}\\x=\frac{412}{7}\text{, }&d=\frac{2}{7}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{11536d-3247}-7}{2\left(7d-2\right)}\text{; }x=-\frac{\sqrt{11536d-3247}+7}{2\left(7d-2\right)}\text{, }&d\neq \frac{2}{7}\text{ and }d\geq \frac{3247}{11536}\\x=\frac{412}{7}\text{, }&d=\frac{2}{7}\end{matrix}\right.$$