Solve ax^2+ax/6xdivx^2-x-3/2x-5=4

$$\frac { a x ^ { 2 } + a x } { 6 x } \div \frac { x ^ { 2 } - x - 3 } { 2 x - 5 } = 4$$

Solve for a

$$a=-\frac{24\left(3+x-x^{2}\right)}{2x^{2}-3x-5}$$

$$x\neq -1\text{ and }x\neq \frac{5}{2}\text{ and }x\neq 0\text{ and }x\neq \frac{\sqrt{13}+1}{2}\text{ and }x\neq \frac{1-\sqrt{13}}{2}$$

Steps for Solving Linear Equation

Divide \frac{ax^{2}+ax}{6x} by \frac{x^{2}-x-3}{2x-5} by multiplying \frac{ax^{2}+ax}{6x} by the reciprocal of \frac{x^{2}-x-3}{2x-5}.

$$\frac{\left(ax^{2}+ax\right)\left(2x-5\right)}{6x\left(x^{2}-x-3\right)}=4$$

Factor the expressions that are not already factored in \frac{\left(ax^{2}+ax\right)\left(2x-5\right)}{6x\left(x^{2}-x-3\right)}.

$$\frac{ax\left(2x-5\right)\left(x+1\right)}{6x\left(x-\left(-\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)}=4$$

Cancel out x in both numerator and denominator.

$$\frac{a\left(2x-5\right)\left(x+1\right)}{6\left(x-\left(-\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)}=4$$

Use the distributive property to multiply a by 2x-5.

$$\frac{\left(2ax-5a\right)\left(x+1\right)}{6\left(x-\left(-\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)}=4$$

Use the distributive property to multiply 2ax-5a by x+1 and combine like terms.

$$\frac{2ax^{2}-3ax-5a}{6\left(x-\left(-\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)}=4$$

To find the opposite of -\frac{1}{2}\sqrt{13}+\frac{1}{2}, find the opposite of each term.

$$\frac{2ax^{2}-3ax-5a}{6\left(x+\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\left(x-\left(\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)}=4$$

To find the opposite of \frac{1}{2}\sqrt{13}+\frac{1}{2}, find the opposite of each term.

$$\frac{2ax^{2}-3ax-5a}{6\left(x+\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\left(x-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)}=4$$

Use the distributive property to multiply 6 by x+\frac{1}{2}\sqrt{13}-\frac{1}{2}.

$$\frac{2ax^{2}-3ax-5a}{\left(6x+3\sqrt{13}-3\right)\left(x-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)}=4$$

Use the distributive property to multiply 6x+3\sqrt{13}-3 by x-\frac{1}{2}\sqrt{13}-\frac{1}{2} and combine like terms.

$$\frac{2ax^{2}-3ax-5a}{6x^{2}-6x-\frac{3}{2}\left(\sqrt{13}\right)^{2}+\frac{3}{2}}=4$$

The square of \sqrt{13} is 13.

$$\frac{2ax^{2}-3ax-5a}{6x^{2}-6x-\frac{3}{2}\times 13+\frac{3}{2}}=4$$

Multiply -\frac{3}{2} and 13 to get -\frac{39}{2}.

$$\frac{2ax^{2}-3ax-5a}{6x^{2}-6x-\frac{39}{2}+\frac{3}{2}}=4$$

Add -\frac{39}{2} and \frac{3}{2} to get -18.

$$\frac{2ax^{2}-3ax-5a}{6x^{2}-6x-18}=4$$

Multiply both sides of the equation by 6\left(x-\left(-\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right).

$$2ax^{2}-3ax-5a=24\left(x-\left(-\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)$$

To find the opposite of -\frac{1}{2}\sqrt{13}+\frac{1}{2}, find the opposite of each term.

$$2ax^{2}-3ax-5a=24\left(x+\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\left(x-\left(\frac{1}{2}\sqrt{13}+\frac{1}{2}\right)\right)$$

To find the opposite of \frac{1}{2}\sqrt{13}+\frac{1}{2}, find the opposite of each term.

$$2ax^{2}-3ax-5a=24\left(x+\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)\left(x-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)$$

Use the distributive property to multiply 24 by x+\frac{1}{2}\sqrt{13}-\frac{1}{2}.

$$2ax^{2}-3ax-5a=\left(24x+12\sqrt{13}-12\right)\left(x-\frac{1}{2}\sqrt{13}-\frac{1}{2}\right)$$

Use the distributive property to multiply 24x+12\sqrt{13}-12 by x-\frac{1}{2}\sqrt{13}-\frac{1}{2} and combine like terms.

$$2ax^{2}-3ax-5a=24x^{2}-24x-6\left(\sqrt{13}\right)^{2}+6$$

The square of \sqrt{13} is 13.

$$2ax^{2}-3ax-5a=24x^{2}-24x-6\times 13+6$$

Multiply -6 and 13 to get -78.

$$2ax^{2}-3ax-5a=24x^{2}-24x-78+6$$

Add -78 and 6 to get -72.

$$2ax^{2}-3ax-5a=24x^{2}-24x-72$$

Combine all terms containing a.

$$\left(2x^{2}-3x-5\right)a=24x^{2}-24x-72$$

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

$$\frac{\left(2x^{2}-3x-5\right)a}{2x^{2}-3x-5}=\frac{24x^{2}-24x-72}{2x^{2}-3x-5}$$

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

$$a=\frac{24x^{2}-24x-72}{2x^{2}-3x-5}$$

Divide 24x^{2}-24x-72 by 2x^{2}-3x-5.

$$a=\frac{24\left(x^{2}-x-3\right)}{2x^{2}-3x-5}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{49a^{2}-1200a+7488}+3a-24}{4\left(a-12\right)}\text{, }&a\neq \frac{72}{5}\text{ and }a\neq 0\text{ and }a\neq 12\\x=\frac{\sqrt{49a^{2}-1200a+7488}+3a-24}{4\left(a-12\right)}\text{, }&a\neq 12\text{ and }a\neq 0\\x=1\text{, }&a=12\end{matrix}\right.$$