Solve (q/p)(x)=frac{sqrt{6-x}}{x^2+5x}

$$( \frac { q } { p } ) ( x ) = \frac { \sqrt { 6 - x } } { x ^ { 2 } + 5 x }$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=\left(6-x\right)^{-\frac{1}{2}}q\left(x+5\right)x^{2}\text{, }&x\neq -5\text{ and }x\neq 0\text{ and }q\neq 0\text{ and }x\neq 6\\p\neq 0\text{, }&x=6\text{ and }q=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable p cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by px\left(x+5\right), the least common multiple of p,x^{2}+5x.

$$x\left(x+5\right)qx=p\sqrt{6-x}$$

Use the distributive property to multiply x by x+5.

$$\left(x^{2}+5x\right)qx=p\sqrt{6-x}$$

Use the distributive property to multiply x^{2}+5x by q.

$$\left(x^{2}q+5xq\right)x=p\sqrt{6-x}$$

Use the distributive property to multiply x^{2}q+5xq by x.

$$qx^{3}+5qx^{2}=p\sqrt{6-x}$$

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

$$p\sqrt{6-x}=qx^{3}+5qx^{2}$$

The equation is in standard form.

$$\sqrt{6-x}p=qx^{3}+5qx^{2}$$

Divide both sides by \sqrt{6-x}.

$$\frac{\sqrt{6-x}p}{\sqrt{6-x}}=\frac{q\left(x+5\right)x^{2}}{\sqrt{6-x}}$$

Dividing by \sqrt{6-x} undoes the multiplication by \sqrt{6-x}.

$$p=\frac{q\left(x+5\right)x^{2}}{\sqrt{6-x}}$$

Divide q\left(5+x\right)x^{2} by \sqrt{6-x}.

$$p=\left(6-x\right)^{-\frac{1}{2}}q\left(x+5\right)x^{2}$$

Variable p cannot be equal to 0.

$$p=\left(6-x\right)^{-\frac{1}{2}}q\left(x+5\right)x^{2}\text{, }p\neq 0$$

Solve for q (complex solution)

$$q=\frac{\sqrt{6-x}p}{\left(x+5\right)x^{2}}$$

$$p\neq 0\text{ and }x\neq -5\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by px\left(x+5\right), the least common multiple of p,x^{2}+5x.

$$x\left(x+5\right)qx=p\sqrt{6-x}$$

Use the distributive property to multiply x by x+5.

$$\left(x^{2}+5x\right)qx=p\sqrt{6-x}$$

Use the distributive property to multiply x^{2}+5x by q.

$$\left(x^{2}q+5xq\right)x=p\sqrt{6-x}$$

Use the distributive property to multiply x^{2}q+5xq by x.

$$qx^{3}+5qx^{2}=p\sqrt{6-x}$$

Combine all terms containing q.

$$\left(x^{3}+5x^{2}\right)q=p\sqrt{6-x}$$

The equation is in standard form.

$$\left(x^{3}+5x^{2}\right)q=\sqrt{6-x}p$$

Divide both sides by 5x^{2}+x^{3}.

$$\frac{\left(x^{3}+5x^{2}\right)q}{x^{3}+5x^{2}}=\frac{\sqrt{6-x}p}{x^{3}+5x^{2}}$$

Dividing by 5x^{2}+x^{3} undoes the multiplication by 5x^{2}+x^{3}.

$$q=\frac{\sqrt{6-x}p}{x^{3}+5x^{2}}$$

Divide p\sqrt{6-x} by 5x^{2}+x^{3}.

$$q=\frac{\sqrt{6-x}p}{\left(x+5\right)x^{2}}$$

Solve for p

$$\left\{\begin{matrix}p=\frac{q\left(x+5\right)x^{2}}{\sqrt{6-x}}\text{, }&x\neq -5\text{ and }x\neq 0\text{ and }q\neq 0\text{ and }x<6\\p\neq 0\text{, }&q=0\text{ and }x=6\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable p cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by px\left(x+5\right), the least common multiple of p,x^{2}+5x.

$$x\left(x+5\right)qx=p\sqrt{6-x}$$

Use the distributive property to multiply x by x+5.

$$\left(x^{2}+5x\right)qx=p\sqrt{6-x}$$

Use the distributive property to multiply x^{2}+5x by q.

$$\left(x^{2}q+5xq\right)x=p\sqrt{6-x}$$

Use the distributive property to multiply x^{2}q+5xq by x.

$$qx^{3}+5qx^{2}=p\sqrt{6-x}$$

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

$$p\sqrt{6-x}=qx^{3}+5qx^{2}$$

The equation is in standard form.

$$\sqrt{6-x}p=qx^{3}+5qx^{2}$$

Divide both sides by \sqrt{6-x}.

$$\frac{\sqrt{6-x}p}{\sqrt{6-x}}=\frac{q\left(x+5\right)x^{2}}{\sqrt{6-x}}$$

Dividing by \sqrt{6-x} undoes the multiplication by \sqrt{6-x}.

$$p=\frac{q\left(x+5\right)x^{2}}{\sqrt{6-x}}$$

Variable p cannot be equal to 0.

$$p=\frac{q\left(x+5\right)x^{2}}{\sqrt{6-x}}\text{, }p\neq 0$$

Solve for q

$$q=\frac{\sqrt{6-x}p}{\left(x+5\right)x^{2}}$$

$$p\neq 0\text{ and }x\leq 6\text{ and }x\neq -5\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by px\left(x+5\right), the least common multiple of p,x^{2}+5x.

$$x\left(x+5\right)qx=p\sqrt{6-x}$$

Use the distributive property to multiply x by x+5.

$$\left(x^{2}+5x\right)qx=p\sqrt{6-x}$$

Use the distributive property to multiply x^{2}+5x by q.

$$\left(x^{2}q+5xq\right)x=p\sqrt{6-x}$$

Use the distributive property to multiply x^{2}q+5xq by x.

$$qx^{3}+5qx^{2}=p\sqrt{6-x}$$

Combine all terms containing q.

$$\left(x^{3}+5x^{2}\right)q=p\sqrt{6-x}$$

The equation is in standard form.

$$\left(x^{3}+5x^{2}\right)q=\sqrt{6-x}p$$

Divide both sides by 5x^{2}+x^{3}.

$$\frac{\left(x^{3}+5x^{2}\right)q}{x^{3}+5x^{2}}=\frac{\sqrt{6-x}p}{x^{3}+5x^{2}}$$

Dividing by 5x^{2}+x^{3} undoes the multiplication by 5x^{2}+x^{3}.

$$q=\frac{\sqrt{6-x}p}{x^{3}+5x^{2}}$$

Divide p\sqrt{6-x} by 5x^{2}+x^{3}.

$$q=\frac{\sqrt{6-x}p}{\left(x+5\right)x^{2}}$$