Solve x^4+9=(x^2+ax+3)(x^2+bx+3)

$$x ^ { 4 } + 9 = ( x ^ { 2 } + a x + 3 ) ( x ^ { 2 } + b x + 3 )$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{bx^{2}+6x+3b}{x^{2}+bx+3}\text{, }&x\neq \frac{-\sqrt{b^{2}-12}-b}{2}\text{ and }\left(x=0\text{ or }b\neq -x-\frac{3}{x}\right)\text{ and }x\neq \frac{\sqrt{b^{2}-12}-b}{2}\\a\in \mathrm{C}\text{, }&\left(b=-\sqrt{6}\text{ and }x=\sqrt{6}\left(\frac{1}{2}-\frac{1}{2}i\right)\right)\text{ or }\left(b=\sqrt{6}\text{ and }x=\sqrt{6}\left(-\frac{1}{2}+\frac{1}{2}i\right)\right)\text{ or }\left(b=-\sqrt{6}\text{ and }x=\sqrt{6}\left(\frac{1}{2}+\frac{1}{2}i\right)\right)\text{ or }\left(b=\sqrt{6}\text{ and }x=\sqrt{6}\left(-\frac{1}{2}-\frac{1}{2}i\right)\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{2}+ax+3 by x^{2}+bx+3 and combine like terms.

$$x^{4}+9=x^{4}+bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9$$

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

$$x^{4}+bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=x^{4}+9$$

Subtract x^{4} from both sides.

$$bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=x^{4}+9-x^{4}$$

Combine x^{4} and -x^{4} to get 0.

$$bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=9$$

Subtract bx^{3} from both sides.

$$6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=9-bx^{3}$$

Subtract 6x^{2} from both sides.

$$ax^{3}+abx^{2}+3ax+3bx+9=9-bx^{3}-6x^{2}$$

Subtract 3bx from both sides.

$$ax^{3}+abx^{2}+3ax+9=9-bx^{3}-6x^{2}-3bx$$

Subtract 9 from both sides.

$$ax^{3}+abx^{2}+3ax=9-bx^{3}-6x^{2}-3bx-9$$

Subtract 9 from 9 to get 0.

$$ax^{3}+abx^{2}+3ax=-bx^{3}-6x^{2}-3bx$$

Combine all terms containing a.

$$\left(x^{3}+bx^{2}+3x\right)a=-bx^{3}-6x^{2}-3bx$$

Divide both sides by x^{3}+bx^{2}+3x.

$$\frac{\left(x^{3}+bx^{2}+3x\right)a}{x^{3}+bx^{2}+3x}=-\frac{x\left(bx^{2}+6x+3b\right)}{x^{3}+bx^{2}+3x}$$

Dividing by x^{3}+bx^{2}+3x undoes the multiplication by x^{3}+bx^{2}+3x.

$$a=-\frac{x\left(bx^{2}+6x+3b\right)}{x^{3}+bx^{2}+3x}$$

Divide -x\left(bx^{2}+6x+3b\right) by x^{3}+bx^{2}+3x.

$$a=-\frac{bx^{2}+6x+3b}{x^{2}+bx+3}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{ax^{2}+6x+3a}{x^{2}+ax+3}\text{, }&x\neq \frac{-\sqrt{a^{2}-12}-a}{2}\text{ and }\left(x=0\text{ or }a\neq -x-\frac{3}{x}\right)\text{ and }x\neq \frac{\sqrt{a^{2}-12}-a}{2}\\b\in \mathrm{C}\text{, }&\left(a=-\sqrt{6}\text{ and }x=\sqrt{6}\left(\frac{1}{2}-\frac{1}{2}i\right)\right)\text{ or }\left(a=\sqrt{6}\text{ and }x=\sqrt{6}\left(-\frac{1}{2}+\frac{1}{2}i\right)\right)\text{ or }\left(a=-\sqrt{6}\text{ and }x=\sqrt{6}\left(\frac{1}{2}+\frac{1}{2}i\right)\right)\text{ or }\left(a=\sqrt{6}\text{ and }x=\sqrt{6}\left(-\frac{1}{2}-\frac{1}{2}i\right)\right)\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{2}+ax+3 by x^{2}+bx+3 and combine like terms.

$$x^{4}+9=x^{4}+bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9$$

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

$$x^{4}+bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=x^{4}+9$$

Subtract x^{4} from both sides.

$$bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=x^{4}+9-x^{4}$$

Combine x^{4} and -x^{4} to get 0.

$$bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=9$$

Subtract 6x^{2} from both sides.

$$bx^{3}+ax^{3}+abx^{2}+3ax+3bx+9=9-6x^{2}$$

Subtract ax^{3} from both sides.

$$bx^{3}+abx^{2}+3ax+3bx+9=9-6x^{2}-ax^{3}$$

Subtract 3ax from both sides.

$$bx^{3}+abx^{2}+3bx+9=9-6x^{2}-ax^{3}-3ax$$

Subtract 9 from both sides.

$$bx^{3}+abx^{2}+3bx=9-6x^{2}-ax^{3}-3ax-9$$

Subtract 9 from 9 to get 0.

$$bx^{3}+abx^{2}+3bx=-6x^{2}-ax^{3}-3ax$$

Combine all terms containing b.

$$\left(x^{3}+ax^{2}+3x\right)b=-6x^{2}-ax^{3}-3ax$$

The equation is in standard form.

$$\left(x^{3}+ax^{2}+3x\right)b=-ax^{3}-6x^{2}-3ax$$

Divide both sides by x^{3}+ax^{2}+3x.

$$\frac{\left(x^{3}+ax^{2}+3x\right)b}{x^{3}+ax^{2}+3x}=-\frac{x\left(ax^{2}+6x+3a\right)}{x^{3}+ax^{2}+3x}$$

Dividing by x^{3}+ax^{2}+3x undoes the multiplication by x^{3}+ax^{2}+3x.

$$b=-\frac{x\left(ax^{2}+6x+3a\right)}{x^{3}+ax^{2}+3x}$$

Divide -x\left(6x+ax^{2}+3a\right) by x^{3}+ax^{2}+3x.

$$b=-\frac{ax^{2}+6x+3a}{x^{2}+ax+3}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{bx^{2}+6x+3b}{x^{2}+bx+3}\text{, }&\left(b>-2\sqrt{3}\text{ and }b\neq -x-\frac{3}{x}\text{ and }x\neq 0\text{ and }|b|<2\sqrt{3}\right)\text{ or }\left(x\neq \frac{\sqrt{b^{2}-12}-b}{2}\text{ and }x\neq \frac{-\sqrt{b^{2}-12}-b}{2}\text{ and }b\neq -x-\frac{3}{x}\text{ and }x\neq 0\text{ and }|b|\geq 2\sqrt{3}\right)\text{ or }x=0\\a\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{2}+ax+3 by x^{2}+bx+3 and combine like terms.

$$x^{4}+9=x^{4}+bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9$$

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

$$x^{4}+bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=x^{4}+9$$

Subtract x^{4} from both sides.

$$bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=x^{4}+9-x^{4}$$

Combine x^{4} and -x^{4} to get 0.

$$bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=9$$

Subtract bx^{3} from both sides.

$$6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=9-bx^{3}$$

Subtract 6x^{2} from both sides.

$$ax^{3}+abx^{2}+3ax+3bx+9=9-bx^{3}-6x^{2}$$

Subtract 3bx from both sides.

$$ax^{3}+abx^{2}+3ax+9=9-bx^{3}-6x^{2}-3bx$$

Subtract 9 from both sides.

$$ax^{3}+abx^{2}+3ax=9-bx^{3}-6x^{2}-3bx-9$$

Subtract 9 from 9 to get 0.

$$ax^{3}+abx^{2}+3ax=-bx^{3}-6x^{2}-3bx$$

Combine all terms containing a.

$$\left(x^{3}+bx^{2}+3x\right)a=-bx^{3}-6x^{2}-3bx$$

Divide both sides by x^{3}+bx^{2}+3x.

$$\frac{\left(x^{3}+bx^{2}+3x\right)a}{x^{3}+bx^{2}+3x}=-\frac{x\left(bx^{2}+6x+3b\right)}{x^{3}+bx^{2}+3x}$$

Dividing by x^{3}+bx^{2}+3x undoes the multiplication by x^{3}+bx^{2}+3x.

$$a=-\frac{x\left(bx^{2}+6x+3b\right)}{x^{3}+bx^{2}+3x}$$

Divide -x\left(bx^{2}+6x+3b\right) by x^{3}+bx^{2}+3x.

$$a=-\frac{bx^{2}+6x+3b}{x^{2}+bx+3}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{ax^{2}+6x+3a}{x^{2}+ax+3}\text{, }&\left(a>-2\sqrt{3}\text{ and }a\neq -x-\frac{3}{x}\text{ and }x\neq 0\text{ and }|a|<2\sqrt{3}\right)\text{ or }\left(x\neq \frac{\sqrt{a^{2}-12}-a}{2}\text{ and }x\neq \frac{-\sqrt{a^{2}-12}-a}{2}\text{ and }a\neq -x-\frac{3}{x}\text{ and }x\neq 0\text{ and }|a|\geq 2\sqrt{3}\right)\text{ or }x=0\\b\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{2}+ax+3 by x^{2}+bx+3 and combine like terms.

$$x^{4}+9=x^{4}+bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9$$

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

$$x^{4}+bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=x^{4}+9$$

Subtract x^{4} from both sides.

$$bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=x^{4}+9-x^{4}$$

Combine x^{4} and -x^{4} to get 0.

$$bx^{3}+6x^{2}+ax^{3}+abx^{2}+3ax+3bx+9=9$$

Subtract 6x^{2} from both sides.

$$bx^{3}+ax^{3}+abx^{2}+3ax+3bx+9=9-6x^{2}$$

Subtract ax^{3} from both sides.

$$bx^{3}+abx^{2}+3ax+3bx+9=9-6x^{2}-ax^{3}$$

Subtract 3ax from both sides.

$$bx^{3}+abx^{2}+3bx+9=9-6x^{2}-ax^{3}-3ax$$

Subtract 9 from both sides.

$$bx^{3}+abx^{2}+3bx=9-6x^{2}-ax^{3}-3ax-9$$

Subtract 9 from 9 to get 0.

$$bx^{3}+abx^{2}+3bx=-6x^{2}-ax^{3}-3ax$$

Combine all terms containing b.

$$\left(x^{3}+ax^{2}+3x\right)b=-6x^{2}-ax^{3}-3ax$$

The equation is in standard form.

$$\left(x^{3}+ax^{2}+3x\right)b=-ax^{3}-6x^{2}-3ax$$

Divide both sides by x^{3}+ax^{2}+3x.

$$\frac{\left(x^{3}+ax^{2}+3x\right)b}{x^{3}+ax^{2}+3x}=-\frac{x\left(ax^{2}+6x+3a\right)}{x^{3}+ax^{2}+3x}$$

Dividing by x^{3}+ax^{2}+3x undoes the multiplication by x^{3}+ax^{2}+3x.

$$b=-\frac{x\left(ax^{2}+6x+3a\right)}{x^{3}+ax^{2}+3x}$$

Divide -x\left(6x+ax^{2}+3a\right) by x^{3}+ax^{2}+3x.

$$b=-\frac{ax^{2}+6x+3a}{x^{2}+ax+3}$$