Solve 10frac{sqrt{3}left(x-1right)a^2}{8xleft(x+2right)}=frac{sqrt{3}left(x+1right)a^2}{8x}

$$10 \frac{ \sqrt{ 3 } \left( x-1 \right) { a }^{ 2 } }{ 8x \left( x+2 \right) } = \frac{ \sqrt{ 3 } \left( x+1 \right) { a }^{ 2 } }{ 8x }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=0\text{, }&x\neq -2\text{ and }x\neq 0\\a\in \mathrm{C}\text{, }&x=3\text{ or }x=4\end{matrix}\right.$$

Steps by Finding Square Root

Multiply both sides of the equation by 8x\left(x+2\right), the least common multiple of 8x\left(x+2\right),8x.

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

Use the distributive property to multiply 10\sqrt{3} by x-1.

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

Use the distributive property to multiply 10\sqrt{3}x-10\sqrt{3} by a^{2}.

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

Use the distributive property to multiply x+2 by \sqrt{3}.

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

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

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

Use the distributive property to multiply \sqrt{3}x^{2}+3x\sqrt{3}+2\sqrt{3} by a^{2}.

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

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

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

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

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

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

$$10\sqrt{3}xa^{2}-10\sqrt{3}a^{2}-\sqrt{3}x^{2}a^{2}-3x\sqrt{3}a^{2}-2\sqrt{3}a^{2}=0$$

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

$$7\sqrt{3}xa^{2}-10\sqrt{3}a^{2}-\sqrt{3}x^{2}a^{2}-2\sqrt{3}a^{2}=0$$

Combine -10\sqrt{3}a^{2} and -2\sqrt{3}a^{2} to get -12\sqrt{3}a^{2}.

$$7\sqrt{3}xa^{2}-12\sqrt{3}a^{2}-\sqrt{3}x^{2}a^{2}=0$$

Combine all terms containing a.

$$\left(7\sqrt{3}x-12\sqrt{3}-\sqrt{3}x^{2}\right)a^{2}=0$$

Dividing by 7\sqrt{3}x-12\sqrt{3}-\sqrt{3}x^{2} undoes the multiplication by 7\sqrt{3}x-12\sqrt{3}-\sqrt{3}x^{2}.

$$a^{2}=\frac{0}{-\sqrt{3}x^{2}+7\sqrt{3}x-12\sqrt{3}}$$

Divide 0 by 7\sqrt{3}x-12\sqrt{3}-\sqrt{3}x^{2}.

$$a^{2}=0$$

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 8x\left(x+2\right), the least common multiple of 8x\left(x+2\right),8x.

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

Use the distributive property to multiply 10\sqrt{3} by x-1.

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

Use the distributive property to multiply 10\sqrt{3}x-10\sqrt{3} by a^{2}.

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

Use the distributive property to multiply x+2 by \sqrt{3}.

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

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

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

Use the distributive property to multiply \sqrt{3}x^{2}+3x\sqrt{3}+2\sqrt{3} by a^{2}.

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

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

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

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

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

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

$$10\sqrt{3}xa^{2}-10\sqrt{3}a^{2}-\sqrt{3}x^{2}a^{2}-3x\sqrt{3}a^{2}-2\sqrt{3}a^{2}=0$$

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

$$7\sqrt{3}xa^{2}-10\sqrt{3}a^{2}-\sqrt{3}x^{2}a^{2}-2\sqrt{3}a^{2}=0$$

Combine -10\sqrt{3}a^{2} and -2\sqrt{3}a^{2} to get -12\sqrt{3}a^{2}.

$$7\sqrt{3}xa^{2}-12\sqrt{3}a^{2}-\sqrt{3}x^{2}a^{2}=0$$

Combine all terms containing a.

$$\left(7\sqrt{3}x-12\sqrt{3}-\sqrt{3}x^{2}\right)a^{2}=0$$

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.

$$\left(-\sqrt{3}x^{2}+7\sqrt{3}x-12\sqrt{3}\right)a^{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7\sqrt{3}x-12\sqrt{3}-\sqrt{3}x^{2} for a, 0 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{0±\sqrt{0^{2}}}{2\left(-\sqrt{3}x^{2}+7\sqrt{3}x-12\sqrt{3}\right)}$$

Take the square root of 0^{2}.

$$a=\frac{0±0}{2\left(-\sqrt{3}x^{2}+7\sqrt{3}x-12\sqrt{3}\right)}$$

Multiply 2 times 7\sqrt{3}x-12\sqrt{3}-\sqrt{3}x^{2}.

$$a=\frac{0}{2\sqrt{3}\left(3-x\right)\left(x-4\right)}$$

Divide 0 by 2\left(-4+x\right)\left(3-x\right)\sqrt{3}.

$$a=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=3\text{; }x=4\text{, }&\text{unconditionally}\\x\in \mathrm{C}\setminus -2,0\text{, }&a=0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a=0\text{, }&x\neq -2\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&x=3\text{ or }x=4\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=3\text{; }x=4\text{, }&\text{unconditionally}\\x\in \mathrm{R}\setminus -2,0\text{, }&a=0\end{matrix}\right.$$