Solve x^2+px+6=(x+q)(x-3)

$$x ^ { 2 } + p x + 6 = ( x + q ) ( x - 3 )$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=\frac{qx-3x-3q-6}{x}\text{, }&x\neq 0\\p\in \mathrm{C}\text{, }&q=-2\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+q by x-3.

$$x^{2}+px+6=x^{2}-3x+qx-3q$$

Subtract x^{2} from both sides.

$$px+6=x^{2}-3x+qx-3q-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$px+6=-3x+qx-3q$$

Subtract 6 from both sides.

$$px=-3x+qx-3q-6$$

The equation is in standard form.

$$xp=qx-3x-3q-6$$

Divide both sides by x.

$$\frac{xp}{x}=\frac{qx-3x-3q-6}{x}$$

Dividing by x undoes the multiplication by x.

$$p=\frac{qx-3x-3q-6}{x}$$

Solve for q (complex solution)

$$\left\{\begin{matrix}q=-\frac{px+3x+6}{3-x}\text{, }&x\neq 3\\q\in \mathrm{C}\text{, }&x=3\text{ and }p=-5\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+q by x-3.

$$x^{2}+px+6=x^{2}-3x+qx-3q$$

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

$$x^{2}-3x+qx-3q=x^{2}+px+6$$

Subtract x^{2} from both sides.

$$-3x+qx-3q=x^{2}+px+6-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-3x+qx-3q=px+6$$

Add 3x to both sides.

$$qx-3q=px+6+3x$$

Combine all terms containing q.

$$\left(x-3\right)q=px+6+3x$$

The equation is in standard form.

$$\left(x-3\right)q=px+3x+6$$

Divide both sides by x-3.

$$\frac{\left(x-3\right)q}{x-3}=\frac{px+3x+6}{x-3}$$

Dividing by x-3 undoes the multiplication by x-3.

$$q=\frac{px+3x+6}{x-3}$$

Solve for p

$$\left\{\begin{matrix}p=\frac{qx-3x-3q-6}{x}\text{, }&x\neq 0\\p\in \mathrm{R}\text{, }&q=-2\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+q by x-3.

$$x^{2}+px+6=x^{2}-3x+qx-3q$$

Subtract x^{2} from both sides.

$$px+6=x^{2}-3x+qx-3q-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$px+6=-3x+qx-3q$$

Subtract 6 from both sides.

$$px=-3x+qx-3q-6$$

The equation is in standard form.

$$xp=qx-3x-3q-6$$

Divide both sides by x.

$$\frac{xp}{x}=\frac{qx-3x-3q-6}{x}$$

Dividing by x undoes the multiplication by x.

$$p=\frac{qx-3x-3q-6}{x}$$

Solve for q

$$\left\{\begin{matrix}q=-\frac{px+3x+6}{3-x}\text{, }&x\neq 3\\q\in \mathrm{R}\text{, }&x=3\text{ and }p=-5\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+q by x-3.

$$x^{2}+px+6=x^{2}-3x+qx-3q$$

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

$$x^{2}-3x+qx-3q=x^{2}+px+6$$

Subtract x^{2} from both sides.

$$-3x+qx-3q=x^{2}+px+6-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$-3x+qx-3q=px+6$$

Add 3x to both sides.

$$qx-3q=px+6+3x$$

Combine all terms containing q.

$$\left(x-3\right)q=px+6+3x$$

The equation is in standard form.

$$\left(x-3\right)q=px+3x+6$$

Divide both sides by x-3.

$$\frac{\left(x-3\right)q}{x-3}=\frac{px+3x+6}{x-3}$$

Dividing by x-3 undoes the multiplication by x-3.

$$q=\frac{px+3x+6}{x-3}$$