Solve frac{sqrt{3}ax}{x+1}(ax-1/2a)=frac{sqrt{3}ax(a+2ax)}{10}

$$\frac { \sqrt { 3 } a x } { x + 1 } ( a x - \frac { 1 } { 2 } a ) = \frac { \sqrt { 3 } a x ( a + 2 a x ) } { 10 }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=0\text{, }&x\neq -1\\a\in \mathrm{C}\text{, }&x=2\text{ or }x=\frac{3}{2}\text{ or }x=0\end{matrix}\right.$$

Steps by Finding Square Root

Multiply both sides of the equation by 10\left(x+1\right), the least common multiple of x+1,2,10.

$$10\sqrt{3}ax\left(ax-\frac{1}{2}a\right)=\left(x+1\right)\sqrt{3}ax\left(a+2ax\right)$$

Use the distributive property to multiply 10\sqrt{3}ax by ax-\frac{1}{2}a.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=\left(x+1\right)\sqrt{3}ax\left(a+2ax\right)$$

Use the distributive property to multiply x+1 by \sqrt{3}.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=\left(x\sqrt{3}+\sqrt{3}\right)ax\left(a+2ax\right)$$

Use the distributive property to multiply x\sqrt{3}+\sqrt{3} by a.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=\left(x\sqrt{3}a+\sqrt{3}a\right)x\left(a+2ax\right)$$

Use the distributive property to multiply x\sqrt{3}a+\sqrt{3}a by x.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=\left(\sqrt{3}ax^{2}+\sqrt{3}ax\right)\left(a+2ax\right)$$

Use the distributive property to multiply \sqrt{3}ax^{2}+\sqrt{3}ax by a+2ax and combine like terms.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=3\sqrt{3}x^{2}a^{2}+2\sqrt{3}a^{2}x^{3}+\sqrt{3}xa^{2}$$

Subtract 3\sqrt{3}x^{2}a^{2} from both sides.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x-3\sqrt{3}x^{2}a^{2}=2\sqrt{3}a^{2}x^{3}+\sqrt{3}xa^{2}$$

Combine 10\sqrt{3}a^{2}x^{2} and -3\sqrt{3}x^{2}a^{2} to get 7\sqrt{3}a^{2}x^{2}.

$$7\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=2\sqrt{3}a^{2}x^{3}+\sqrt{3}xa^{2}$$

Subtract 2\sqrt{3}a^{2}x^{3} from both sides.

$$7\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x-2\sqrt{3}a^{2}x^{3}=\sqrt{3}xa^{2}$$

Subtract \sqrt{3}xa^{2} from both sides.

$$7\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x-2\sqrt{3}a^{2}x^{3}-\sqrt{3}xa^{2}=0$$

Combine -5\sqrt{3}a^{2}x and -\sqrt{3}xa^{2} to get -6\sqrt{3}a^{2}x.

$$7\sqrt{3}a^{2}x^{2}-6\sqrt{3}a^{2}x-2\sqrt{3}a^{2}x^{3}=0$$

Combine all terms containing a.

$$\left(7\sqrt{3}x^{2}-6\sqrt{3}x-2\sqrt{3}x^{3}\right)a^{2}=0$$

Dividing by 7\sqrt{3}x^{2}-6\sqrt{3}x-2\sqrt{3}x^{3} undoes the multiplication by 7\sqrt{3}x^{2}-6\sqrt{3}x-2\sqrt{3}x^{3}.

$$a^{2}=\frac{0}{-2\sqrt{3}x^{3}+7\sqrt{3}x^{2}-6\sqrt{3}x}$$

Divide 0 by 7\sqrt{3}x^{2}-6\sqrt{3}x-2\sqrt{3}x^{3}.

$$a^{2}=0$$

Take the square root of both sides of the equation.

$$a=0 a=0$$

The equation is now solved. Solutions are the same.

$$a=0$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 10\left(x+1\right), the least common multiple of x+1,2,10.

$$10\sqrt{3}ax\left(ax-\frac{1}{2}a\right)=\left(x+1\right)\sqrt{3}ax\left(a+2ax\right)$$

Use the distributive property to multiply 10\sqrt{3}ax by ax-\frac{1}{2}a.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=\left(x+1\right)\sqrt{3}ax\left(a+2ax\right)$$

Use the distributive property to multiply x+1 by \sqrt{3}.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=\left(x\sqrt{3}+\sqrt{3}\right)ax\left(a+2ax\right)$$

Use the distributive property to multiply x\sqrt{3}+\sqrt{3} by a.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=\left(x\sqrt{3}a+\sqrt{3}a\right)x\left(a+2ax\right)$$

Use the distributive property to multiply x\sqrt{3}a+\sqrt{3}a by x.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=\left(\sqrt{3}ax^{2}+\sqrt{3}ax\right)\left(a+2ax\right)$$

Use the distributive property to multiply \sqrt{3}ax^{2}+\sqrt{3}ax by a+2ax and combine like terms.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=3\sqrt{3}x^{2}a^{2}+2\sqrt{3}a^{2}x^{3}+\sqrt{3}xa^{2}$$

Subtract 3\sqrt{3}x^{2}a^{2} from both sides.

$$10\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x-3\sqrt{3}x^{2}a^{2}=2\sqrt{3}a^{2}x^{3}+\sqrt{3}xa^{2}$$

Combine 10\sqrt{3}a^{2}x^{2} and -3\sqrt{3}x^{2}a^{2} to get 7\sqrt{3}a^{2}x^{2}.

$$7\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x=2\sqrt{3}a^{2}x^{3}+\sqrt{3}xa^{2}$$

Subtract 2\sqrt{3}a^{2}x^{3} from both sides.

$$7\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x-2\sqrt{3}a^{2}x^{3}=\sqrt{3}xa^{2}$$

Subtract \sqrt{3}xa^{2} from both sides.

$$7\sqrt{3}a^{2}x^{2}-5\sqrt{3}a^{2}x-2\sqrt{3}a^{2}x^{3}-\sqrt{3}xa^{2}=0$$

Combine -5\sqrt{3}a^{2}x and -\sqrt{3}xa^{2} to get -6\sqrt{3}a^{2}x.

$$7\sqrt{3}a^{2}x^{2}-6\sqrt{3}a^{2}x-2\sqrt{3}a^{2}x^{3}=0$$

Combine all terms containing a.

$$\left(7\sqrt{3}x^{2}-6\sqrt{3}x-2\sqrt{3}x^{3}\right)a^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$\left(-2\sqrt{3}x^{3}+7\sqrt{3}x^{2}-6\sqrt{3}x\right)a^{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7\sqrt{3}x^{2}-6\sqrt{3}x-2\sqrt{3}x^{3} for a, 0 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{0±\sqrt{0^{2}}}{2\left(-2\sqrt{3}x^{3}+7\sqrt{3}x^{2}-6\sqrt{3}x\right)}$$

Take the square root of 0^{2}.

$$a=\frac{0±0}{2\left(-2\sqrt{3}x^{3}+7\sqrt{3}x^{2}-6\sqrt{3}x\right)}$$

Multiply 2 times 7\sqrt{3}x^{2}-6\sqrt{3}x-2\sqrt{3}x^{3}.

$$a=\frac{0}{2\sqrt{3}x\left(2-x\right)\left(2x-3\right)}$$

Divide 0 by 2x\left(-3+2x\right)\left(2-x\right)\sqrt{3}.

$$a=0$$

Solve for a

$$\left\{\begin{matrix}a=0\text{, }&x\neq -1\\a\in \mathrm{R}\text{, }&x=2\text{ or }x=\frac{3}{2}\text{ or }x=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{3}{2}\text{; }x=0\text{; }x=2\text{, }&\text{unconditionally}\\x\neq -1\text{, }&a=0\end{matrix}\right.$$