Solve frac{sqrt{3}Kt^2}{-2}+t^3+frac{sqrt{3}t}{2}-frac{3sqrt{3}k}{2}-3k-3t-2sqrt{3}t+25kt^2-6sqrt{3}k-4k=0

$$\frac { \sqrt { 3 } K t ^ { 2 } } { - 2 } + t ^ { 3 } + \frac { \sqrt { 3 } t } { 2 } - \frac { 3 \sqrt { 3 } k } { 2 } - 3 k - 3 t - 2 \sqrt { 3 } t + 25 k t ^ { 2 } - 6 \sqrt { 3 } k - 4 k = 0$$

Solve for K

$$\left\{\begin{matrix}K=\frac{50\sqrt{3}kt^{2}+2\sqrt{3}t^{3}-6\sqrt{3}t-14\sqrt{3}k-9t-45k}{3t^{2}}\text{, }&t\neq 0\\K\in \mathrm{R}\text{, }&t=0\text{ and }k=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2, the least common multiple of -2,2.

$$2\left(\frac{\sqrt{3}Kt^{2}}{-2}+t^{3}+\frac{\sqrt{3}t}{2}-\frac{3\sqrt{3}k}{2}-3k-3t-2\sqrt{3}t+25kt^{2}-6\sqrt{3}k\right)-8k=0$$

Since \frac{\sqrt{3}t}{2} and \frac{3\sqrt{3}k}{2} have the same denominator, subtract them by subtracting their numerators.

$$2\left(\frac{\sqrt{3}Kt^{2}}{-2}+t^{3}+\frac{\sqrt{3}t-3\sqrt{3}k}{2}-3k-3t-2\sqrt{3}t+25kt^{2}-6\sqrt{3}k\right)-8k=0$$

Add 8k to both sides. Anything plus zero gives itself.

$$2\left(\frac{\sqrt{3}Kt^{2}}{-2}+t^{3}+\frac{\sqrt{3}t-3\sqrt{3}k}{2}-3k-3t-2\sqrt{3}t+25kt^{2}-6\sqrt{3}k\right)=8k$$

Multiply both sides of the equation by 2, the least common multiple of -2,2.

$$4\left(\frac{\sqrt{3}Kt^{2}}{-2}+t^{3}+\frac{\sqrt{3}t-3\sqrt{3}k}{2}-3k-3t-2\sqrt{3}t+25kt^{2}-6\sqrt{3}k\right)=16k$$

Multiply both sides of the equation by 2, the least common multiple of -2,2.

$$8\left(\frac{\sqrt{3}Kt^{2}}{-2}+t^{3}+\frac{\sqrt{3}t-3\sqrt{3}k}{2}-3k-3t-2\sqrt{3}t+25kt^{2}-6\sqrt{3}k\right)=32k$$

Use the distributive property to multiply 8 by \frac{\sqrt{3}Kt^{2}}{-2}+t^{3}+\frac{\sqrt{3}t-3\sqrt{3}k}{2}-3k-3t-2\sqrt{3}t+25kt^{2}-6\sqrt{3}k.

$$8\times \frac{\sqrt{3}Kt^{2}}{-2}+8t^{3}+8\times \frac{\sqrt{3}t-3\sqrt{3}k}{2}-24k-24t-16\sqrt{3}t+200kt^{2}-48\sqrt{3}k=32k$$

Cancel out -2, the greatest common factor in 8 and -2.

$$4\sqrt{3}Kt^{2}+8t^{3}+8\times \frac{\sqrt{3}t-3\sqrt{3}k}{2}-24k-24t-16\sqrt{3}t+200kt^{2}-48\sqrt{3}k=32k$$

Cancel out 2, the greatest common factor in 8 and 2.

$$4\sqrt{3}Kt^{2}+8t^{3}+4\left(\sqrt{3}t-3\sqrt{3}k\right)-24k-24t-16\sqrt{3}t+200kt^{2}-48\sqrt{3}k=32k$$

Use the distributive property to multiply 4 by \sqrt{3}t-3\sqrt{3}k.

$$4\sqrt{3}Kt^{2}+8t^{3}+4\sqrt{3}t-12\sqrt{3}k-24k-24t-16\sqrt{3}t+200kt^{2}-48\sqrt{3}k=32k$$

Combine 4\sqrt{3}t and -16\sqrt{3}t to get -12\sqrt{3}t.

$$4\sqrt{3}Kt^{2}+8t^{3}-12\sqrt{3}t-12\sqrt{3}k-24k-24t+200kt^{2}-48\sqrt{3}k=32k$$

Combine -12\sqrt{3}k and -48\sqrt{3}k to get -60\sqrt{3}k.

$$4\sqrt{3}Kt^{2}+8t^{3}-12\sqrt{3}t-60\sqrt{3}k-24k-24t+200kt^{2}=32k$$

Subtract 8t^{3} from both sides.

$$4\sqrt{3}Kt^{2}-12\sqrt{3}t-60\sqrt{3}k-24k-24t+200kt^{2}=32k-8t^{3}$$

Add 12\sqrt{3}t to both sides.

$$4\sqrt{3}Kt^{2}-60\sqrt{3}k-24k-24t+200kt^{2}=32k-8t^{3}+12\sqrt{3}t$$

Add 60\sqrt{3}k to both sides.

$$4\sqrt{3}Kt^{2}-24k-24t+200kt^{2}=32k-8t^{3}+12\sqrt{3}t+60\sqrt{3}k$$

Add 24k to both sides.

$$4\sqrt{3}Kt^{2}-24t+200kt^{2}=32k-8t^{3}+12\sqrt{3}t+60\sqrt{3}k+24k$$

Combine 32k and 24k to get 56k.

$$4\sqrt{3}Kt^{2}-24t+200kt^{2}=56k-8t^{3}+12\sqrt{3}t+60\sqrt{3}k$$

Add 24t to both sides.

$$4\sqrt{3}Kt^{2}+200kt^{2}=56k-8t^{3}+12\sqrt{3}t+60\sqrt{3}k+24t$$

Subtract 200kt^{2} from both sides.

$$4\sqrt{3}Kt^{2}=56k-8t^{3}+12\sqrt{3}t+60\sqrt{3}k+24t-200kt^{2}$$

The equation is in standard form.

$$4\sqrt{3}t^{2}K=-200kt^{2}-8t^{3}+12\sqrt{3}t+60\sqrt{3}k+24t+56k$$

Divide both sides by 4\sqrt{3}t^{2}.

$$\frac{4\sqrt{3}t^{2}K}{4\sqrt{3}t^{2}}=\frac{-200kt^{2}-8t^{3}+12\sqrt{3}t+60\sqrt{3}k+24t+56k}{4\sqrt{3}t^{2}}$$

Dividing by 4\sqrt{3}t^{2} undoes the multiplication by 4\sqrt{3}t^{2}.

$$K=\frac{-200kt^{2}-8t^{3}+12\sqrt{3}t+60\sqrt{3}k+24t+56k}{4\sqrt{3}t^{2}}$$

Divide 56k-8t^{3}+12t\sqrt{3}+60\sqrt{3}k+24t-200kt^{2} by 4\sqrt{3}t^{2}.

$$K=\frac{\sqrt{3}\left(-50kt^{2}-2t^{3}+3\sqrt{3}t+15\sqrt{3}k+6t+14k\right)}{3t^{2}}$$

Solve for k

$$\left\{\begin{matrix}k=-\frac{t\left(-\sqrt{3}Kt+2t^{2}-3\sqrt{3}-6\right)}{50t^{2}-15\sqrt{3}-14}\text{, }&|t|\neq \frac{\sqrt{30\sqrt{3}+28}}{10}\\k\in \mathrm{R}\text{, }&\left(t=-\frac{\sqrt{30\sqrt{3}+28}}{10}\text{ and }K=-\left(\frac{\sqrt{90\sqrt{3}+84}}{15}-\frac{5\sqrt{140826\sqrt{3}+235668}}{479}\right)\right)\text{ or }\left(t=\frac{\sqrt{30\sqrt{3}+28}}{10}\text{ and }K=\frac{\sqrt{90\sqrt{3}+84}}{15}-\frac{5\sqrt{140826\sqrt{3}+235668}}{479}\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2, the least common multiple of -2,2.

$$2\left(\frac{\sqrt{3}Kt^{2}}{-2}+t^{3}+\frac{\sqrt{3}t}{2}-\frac{3\sqrt{3}k}{2}-3k-3t-2\sqrt{3}t+25kt^{2}-6\sqrt{3}k\right)-8k=0$$

Since \frac{\sqrt{3}t}{2} and \frac{3\sqrt{3}k}{2} have the same denominator, subtract them by subtracting their numerators.

$$2\left(\frac{\sqrt{3}Kt^{2}}{-2}+t^{3}+\frac{\sqrt{3}t-3\sqrt{3}k}{2}-3k-3t-2\sqrt{3}t+25kt^{2}-6\sqrt{3}k\right)-8k=0$$

Multiply both sides of the equation by 2, the least common multiple of -2,2.

$$4\left(\frac{\sqrt{3}Kt^{2}}{-2}+t^{3}+\frac{\sqrt{3}t-3\sqrt{3}k}{2}-3k-3t-2\sqrt{3}t+25kt^{2}-6\sqrt{3}k\right)-16k=0$$

Multiply both sides of the equation by 2, the least common multiple of -2,2.

$$8\left(\frac{\sqrt{3}Kt^{2}}{-2}+t^{3}+\frac{\sqrt{3}t-3\sqrt{3}k}{2}-3k-3t-2\sqrt{3}t+25kt^{2}-6\sqrt{3}k\right)-32k=0$$

Use the distributive property to multiply 8 by \frac{\sqrt{3}Kt^{2}}{-2}+t^{3}+\frac{\sqrt{3}t-3\sqrt{3}k}{2}-3k-3t-2\sqrt{3}t+25kt^{2}-6\sqrt{3}k.

$$8\times \frac{\sqrt{3}Kt^{2}}{-2}+8t^{3}+8\times \frac{\sqrt{3}t-3\sqrt{3}k}{2}-24k-24t-16\sqrt{3}t+200t^{2}k-48\sqrt{3}k-32k=0$$

Cancel out -2, the greatest common factor in 8 and -2.

$$4\sqrt{3}Kt^{2}+8t^{3}+8\times \frac{\sqrt{3}t-3\sqrt{3}k}{2}-24k-24t-16\sqrt{3}t+200t^{2}k-48\sqrt{3}k-32k=0$$

Cancel out 2, the greatest common factor in 8 and 2.

$$4\sqrt{3}Kt^{2}+8t^{3}+4\left(\sqrt{3}t-3\sqrt{3}k\right)-24k-24t-16\sqrt{3}t+200t^{2}k-48\sqrt{3}k-32k=0$$

Use the distributive property to multiply 4 by \sqrt{3}t-3\sqrt{3}k.

$$4\sqrt{3}Kt^{2}+8t^{3}+4\sqrt{3}t-12\sqrt{3}k-24k-24t-16\sqrt{3}t+200t^{2}k-48\sqrt{3}k-32k=0$$

Combine 4\sqrt{3}t and -16\sqrt{3}t to get -12\sqrt{3}t.

$$4\sqrt{3}Kt^{2}+8t^{3}-12\sqrt{3}t-12\sqrt{3}k-24k-24t+200t^{2}k-48\sqrt{3}k-32k=0$$

Combine -12\sqrt{3}k and -48\sqrt{3}k to get -60\sqrt{3}k.

$$4\sqrt{3}Kt^{2}+8t^{3}-12\sqrt{3}t-60\sqrt{3}k-24k-24t+200t^{2}k-32k=0$$

Combine -24k and -32k to get -56k.

$$4\sqrt{3}Kt^{2}+8t^{3}-12\sqrt{3}t-60\sqrt{3}k-56k-24t+200t^{2}k=0$$

Subtract 4\sqrt{3}Kt^{2} from both sides. Anything subtracted from zero gives its negation.

$$8t^{3}-12\sqrt{3}t-60\sqrt{3}k-56k-24t+200t^{2}k=-4\sqrt{3}Kt^{2}$$

Subtract 8t^{3} from both sides.

$$-12\sqrt{3}t-60\sqrt{3}k-56k-24t+200t^{2}k=-4\sqrt{3}Kt^{2}-8t^{3}$$

Add 12\sqrt{3}t to both sides.

$$-60\sqrt{3}k-56k-24t+200t^{2}k=-4\sqrt{3}Kt^{2}-8t^{3}+12\sqrt{3}t$$

Add 24t to both sides.

$$-60\sqrt{3}k-56k+200t^{2}k=-4\sqrt{3}Kt^{2}-8t^{3}+12\sqrt{3}t+24t$$

Combine all terms containing k.

$$\left(-60\sqrt{3}-56+200t^{2}\right)k=-4\sqrt{3}Kt^{2}-8t^{3}+12\sqrt{3}t+24t$$

The equation is in standard form.

$$\left(200t^{2}-60\sqrt{3}-56\right)k=-4\sqrt{3}Kt^{2}-8t^{3}+12\sqrt{3}t+24t$$

Divide both sides by -60\sqrt{3}-56+200t^{2}.

$$\frac{\left(200t^{2}-60\sqrt{3}-56\right)k}{200t^{2}-60\sqrt{3}-56}=\frac{4t\left(-\sqrt{3}Kt-2t^{2}+3\sqrt{3}+6\right)}{200t^{2}-60\sqrt{3}-56}$$

Dividing by -60\sqrt{3}-56+200t^{2} undoes the multiplication by -60\sqrt{3}-56+200t^{2}.

$$k=\frac{4t\left(-\sqrt{3}Kt-2t^{2}+3\sqrt{3}+6\right)}{200t^{2}-60\sqrt{3}-56}$$

Divide 4t\left(-\sqrt{3}Kt-2t^{2}+3\sqrt{3}+6\right) by -60\sqrt{3}-56+200t^{2}.

$$k=\frac{t\left(-\sqrt{3}Kt-2t^{2}+3\sqrt{3}+6\right)}{50t^{2}-15\sqrt{3}-14}$$