Solve x-95/2+17/8-9sqrt{2+17/2}=frac{4-sqrt{2}}{-3-sqrt{2}}

$$\frac { x - \frac { 95 } { 2 } + 17 } { \frac { 8 - 9 \sqrt { 2 } + 17 } { 2 } } = \frac { 4 - \sqrt { 2 } } { - 3 - \sqrt { 2 } }$$

Solve for x

$$x = \frac{43 \sqrt{2} - 7}{2} \approx 26.905591591$$

Steps for Solving Linear Equation

Convert 17 to fraction \frac{34}{2}.

$$\frac{x-\frac{95}{2}+\frac{34}{2}}{\frac{8-9\sqrt{2}+17}{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Since -\frac{95}{2} and \frac{34}{2} have the same denominator, add them by adding their numerators.

$$\frac{x+\frac{-95+34}{2}}{\frac{8-9\sqrt{2}+17}{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Add -95 and 34 to get -61.

$$\frac{x-\frac{61}{2}}{\frac{8-9\sqrt{2}+17}{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Add 8 and 17 to get 25.

$$\frac{x-\frac{61}{2}}{\frac{25-9\sqrt{2}}{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Divide x-\frac{61}{2} by \frac{25-9\sqrt{2}}{2} by multiplying x-\frac{61}{2} by the reciprocal of \frac{25-9\sqrt{2}}{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2}{25-9\sqrt{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Rationalize the denominator of \frac{\left(x-\frac{61}{2}\right)\times 2}{25-9\sqrt{2}} by multiplying numerator and denominator by 25+9\sqrt{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{\left(25-9\sqrt{2}\right)\left(25+9\sqrt{2}\right)}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Consider \left(25-9\sqrt{2}\right)\left(25+9\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{25^{2}-\left(-9\sqrt{2}\right)^{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Calculate 25 to the power of 2 and get 625.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{625-\left(-9\sqrt{2}\right)^{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Expand \left(-9\sqrt{2}\right)^{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{625-\left(-9\right)^{2}\left(\sqrt{2}\right)^{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Calculate -9 to the power of 2 and get 81.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{625-81\left(\sqrt{2}\right)^{2}}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{625-81\times 2}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Multiply 81 and 2 to get 162.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{625-162}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Subtract 162 from 625 to get 463.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{463}=\frac{4-\sqrt{2}}{-3-\sqrt{2}}$$

Rationalize the denominator of \frac{4-\sqrt{2}}{-3-\sqrt{2}} by multiplying numerator and denominator by -3+\sqrt{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{\left(-3-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}$$

Consider \left(-3-\sqrt{2}\right)\left(-3+\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{\left(-3\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square -3. Square \sqrt{2}.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{9-2}$$

Subtract 2 from 9 to get 7.

$$\frac{\left(x-\frac{61}{2}\right)\times 2\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{7}$$

Use the distributive property to multiply x-\frac{61}{2} by 2.

$$\frac{\left(2x-\frac{61}{2}\times 2\right)\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{7}$$

Cancel out 2 and 2.

$$\frac{\left(2x-61\right)\left(25+9\sqrt{2}\right)}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{7}$$

Apply the distributive property by multiplying each term of 2x-61 by each term of 25+9\sqrt{2}.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=\frac{\left(4-\sqrt{2}\right)\left(-3+\sqrt{2}\right)}{7}$$

Apply the distributive property by multiplying each term of 4-\sqrt{2} by each term of -3+\sqrt{2}.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=\frac{-12+4\sqrt{2}+3\sqrt{2}-\left(\sqrt{2}\right)^{2}}{7}$$

Combine 4\sqrt{2} and 3\sqrt{2} to get 7\sqrt{2}.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=\frac{-12+7\sqrt{2}-\left(\sqrt{2}\right)^{2}}{7}$$

The square of \sqrt{2} is 2.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=\frac{-12+7\sqrt{2}-2}{7}$$

Subtract 2 from -12 to get -14.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=\frac{-14+7\sqrt{2}}{7}$$

Divide each term of -14+7\sqrt{2} by 7 to get -2+\sqrt{2}.

$$\frac{50x+18\sqrt{2}x-1525-549\sqrt{2}}{463}=-2+\sqrt{2}$$

Multiply both sides of the equation by 463.

$$50x+18\sqrt{2}x-1525-549\sqrt{2}=-926+463\sqrt{2}$$

Add 1525 to both sides.

$$50x+18\sqrt{2}x-549\sqrt{2}=-926+463\sqrt{2}+1525$$

Add -926 and 1525 to get 599.

$$50x+18\sqrt{2}x-549\sqrt{2}=599+463\sqrt{2}$$

Add 549\sqrt{2} to both sides.

$$50x+18\sqrt{2}x=599+463\sqrt{2}+549\sqrt{2}$$

Combine 463\sqrt{2} and 549\sqrt{2} to get 1012\sqrt{2}.

$$50x+18\sqrt{2}x=599+1012\sqrt{2}$$

Combine all terms containing x.

$$\left(50+18\sqrt{2}\right)x=599+1012\sqrt{2}$$

The equation is in standard form.

$$\left(18\sqrt{2}+50\right)x=1012\sqrt{2}+599$$

Divide both sides by 50+18\sqrt{2}.

$$\frac{\left(18\sqrt{2}+50\right)x}{18\sqrt{2}+50}=\frac{1012\sqrt{2}+599}{18\sqrt{2}+50}$$

Dividing by 50+18\sqrt{2} undoes the multiplication by 50+18\sqrt{2}.

$$x=\frac{1012\sqrt{2}+599}{18\sqrt{2}+50}$$

Divide 599+1012\sqrt{2} by 50+18\sqrt{2}.

$$x=\frac{43\sqrt{2}-7}{2}$$