Solve ax^3+bx^2+a+d=0

$$a x ^ { 3 } + b x ^ { 2 } + a + d = 0$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{bx^{2}+d}{x^{3}+1}\text{, }&x\neq -1\\a\in \mathrm{R}\text{, }&b=-d\text{ and }x=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract bx^{2} from both sides. Anything subtracted from zero gives its negation.

$$ax^{3}+a+d=-bx^{2}$$

Subtract d from both sides.

$$ax^{3}+a=-bx^{2}-d$$

Combine all terms containing a.

$$\left(x^{3}+1\right)a=-bx^{2}-d$$

Divide both sides by x^{3}+1.

$$\frac{\left(x^{3}+1\right)a}{x^{3}+1}=\frac{-bx^{2}-d}{x^{3}+1}$$

Dividing by x^{3}+1 undoes the multiplication by x^{3}+1.

$$a=\frac{-bx^{2}-d}{x^{3}+1}$$

Divide -bx^{2}-d by x^{3}+1.

$$a=-\frac{bx^{2}+d}{\left(x+1\right)\left(x^{2}-x+1\right)}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{ax^{3}+a+d}{x^{2}}\text{, }&x\neq 0\\b\in \mathrm{R}\text{, }&a=-d\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract ax^{3} from both sides. Anything subtracted from zero gives its negation.

$$bx^{2}+a+d=-ax^{3}$$

Subtract a from both sides.

$$bx^{2}+d=-ax^{3}-a$$

Subtract d from both sides.

$$bx^{2}=-ax^{3}-a-d$$

The equation is in standard form.

$$x^{2}b=-ax^{3}-a-d$$

Divide both sides by x^{2}.

$$\frac{x^{2}b}{x^{2}}=\frac{-ax^{3}-a-d}{x^{2}}$$

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

$$b=\frac{-ax^{3}-a-d}{x^{2}}$$

Divide -ax^{3}-a-d by x^{2}.

$$b=-\frac{ax^{3}+a+d}{x^{2}}$$