Solve 3/x-2+2/-25-3sqrt{41/2+1}=17/5

$$\frac{ 3 }{ x-2 } + \frac{ 2 }{ \frac{ -25-3 \sqrt{ 41 } }{ 2 } +1 } = \frac{ 17 }{ 5 }$$

Solve for x

$$x = \frac{5 \sqrt{41} + 957}{346} \approx 2.858426651$$

Steps for Solving Linear Equation

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-2\right), the least common multiple of x-2,5.

$$5\times 3-\frac{1}{16}\left(x-2\right)\left(-3\times 41^{\frac{1}{2}}+23\right)\times 2=17\left(x-2\right)$$

Multiply 5 and 3 to get 15.

$$15-\frac{1}{16}\left(x-2\right)\left(-3\times 41^{\frac{1}{2}}+23\right)\times 2=17\left(x-2\right)$$

Multiply -\frac{1}{16} and 2 to get -\frac{1}{8}.

$$15-\frac{1}{8}\left(x-2\right)\left(-3\times 41^{\frac{1}{2}}+23\right)=17\left(x-2\right)$$

Use the distributive property to multiply -\frac{1}{8} by x-2.

$$15+\left(-\frac{1}{8}x+\frac{1}{4}\right)\left(-3\times 41^{\frac{1}{2}}+23\right)=17\left(x-2\right)$$

Use the distributive property to multiply -\frac{1}{8}x+\frac{1}{4} by -3\times 41^{\frac{1}{2}}+23.

$$15+\frac{3}{8}x\times 41^{\frac{1}{2}}-\frac{23}{8}x-\frac{3}{4}\times 41^{\frac{1}{2}}+\frac{23}{4}=17\left(x-2\right)$$

Add 15 and \frac{23}{4} to get \frac{83}{4}.

$$\frac{83}{4}+\frac{3}{8}x\times 41^{\frac{1}{2}}-\frac{23}{8}x-\frac{3}{4}\times 41^{\frac{1}{2}}=17\left(x-2\right)$$

Use the distributive property to multiply 17 by x-2.

$$\frac{83}{4}+\frac{3}{8}x\times 41^{\frac{1}{2}}-\frac{23}{8}x-\frac{3}{4}\times 41^{\frac{1}{2}}=17x-34$$

Subtract 17x from both sides.

$$\frac{83}{4}+\frac{3}{8}x\times 41^{\frac{1}{2}}-\frac{23}{8}x-\frac{3}{4}\times 41^{\frac{1}{2}}-17x=-34$$

Combine -\frac{23}{8}x and -17x to get -\frac{159}{8}x.

$$\frac{83}{4}+\frac{3}{8}x\times 41^{\frac{1}{2}}-\frac{159}{8}x-\frac{3}{4}\times 41^{\frac{1}{2}}=-34$$

Subtract \frac{83}{4} from both sides.

$$\frac{3}{8}x\times 41^{\frac{1}{2}}-\frac{159}{8}x-\frac{3}{4}\times 41^{\frac{1}{2}}=-34-\frac{83}{4}$$

Subtract \frac{83}{4} from -34 to get -\frac{219}{4}.

$$\frac{3}{8}x\times 41^{\frac{1}{2}}-\frac{159}{8}x-\frac{3}{4}\times 41^{\frac{1}{2}}=-\frac{219}{4}$$

Add \frac{3}{4}\times 41^{\frac{1}{2}} to both sides.

$$\frac{3}{8}x\times 41^{\frac{1}{2}}-\frac{159}{8}x=-\frac{219}{4}+\frac{3}{4}\times 41^{\frac{1}{2}}$$

Reorder the terms.

$$\frac{3}{8}\sqrt{41}x-\frac{159}{8}x=\frac{3}{4}\sqrt{41}-\frac{219}{4}$$

Combine all terms containing x.

$$\left(\frac{3}{8}\sqrt{41}-\frac{159}{8}\right)x=\frac{3}{4}\sqrt{41}-\frac{219}{4}$$

The equation is in standard form.

$$\frac{3\sqrt{41}-159}{8}x=\frac{3\sqrt{41}-219}{4}$$

Divide both sides by \frac{3}{8}\sqrt{41}-\frac{159}{8}.

$$\frac{8\times \frac{3\sqrt{41}-159}{8}x}{3\sqrt{41}-159}=\frac{3\sqrt{41}-219}{4\times \frac{3\sqrt{41}-159}{8}}$$

Dividing by \frac{3}{8}\sqrt{41}-\frac{159}{8} undoes the multiplication by \frac{3}{8}\sqrt{41}-\frac{159}{8}.

$$x=\frac{3\sqrt{41}-219}{4\times \frac{3\sqrt{41}-159}{8}}$$

Divide \frac{3\sqrt{41}-219}{4} by \frac{3}{8}\sqrt{41}-\frac{159}{8}.

$$x=\frac{5\sqrt{41}+957}{346}$$