Solve (frac{3+sqrt{17}}{2})+(frac{3+sqrt{17}}{2}-3)^2

$$( \frac { 3 + \sqrt { 17 } } { 2 } ) + ( \frac { 3 + \sqrt { 17 } } { 2 } - 3 ) ^ { 2 }$$

Evaluate

$$8-\sqrt{17}\approx 3.876894374$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{2}{2}.

$$\frac{3+\sqrt{17}}{2}+\left(\frac{3+\sqrt{17}}{2}-\frac{3\times 2}{2}\right)^{2}$$

Since \frac{3+\sqrt{17}}{2} and \frac{3\times 2}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3+\sqrt{17}}{2}+\left(\frac{3+\sqrt{17}-3\times 2}{2}\right)^{2}$$

Do the multiplications in 3+\sqrt{17}-3\times 2.

$$\frac{3+\sqrt{17}}{2}+\left(\frac{3+\sqrt{17}-6}{2}\right)^{2}$$

Do the calculations in 3+\sqrt{17}-6.

$$\frac{3+\sqrt{17}}{2}+\left(\frac{-3+\sqrt{17}}{2}\right)^{2}$$

To raise \frac{-3+\sqrt{17}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{3+\sqrt{17}}{2}+\frac{\left(-3+\sqrt{17}\right)^{2}}{2^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 2^{2} is 4. Multiply \frac{3+\sqrt{17}}{2} times \frac{2}{2}.

$$\frac{2\left(3+\sqrt{17}\right)}{4}+\frac{\left(-3+\sqrt{17}\right)^{2}}{4}$$

Since \frac{2\left(3+\sqrt{17}\right)}{4} and \frac{\left(-3+\sqrt{17}\right)^{2}}{4} have the same denominator, add them by adding their numerators.

$$\frac{2\left(3+\sqrt{17}\right)+\left(-3+\sqrt{17}\right)^{2}}{4}$$

Do the multiplications in 2\left(3+\sqrt{17}\right)+\left(-3+\sqrt{17}\right)^{2}.

$$\frac{6+2\sqrt{17}+9-6\sqrt{17}+\left(\sqrt{17}\right)^{2}}{4}$$

Do the calculations in 6+2\sqrt{17}+9-6\sqrt{17}+\left(\sqrt{17}\right)^{2}.

$$\frac{32-4\sqrt{17}}{4}$$

Divide each term of 32-4\sqrt{17} by 4 to get 8-\sqrt{17}.

$$8-\sqrt{17}$$

Factor

$$8-\sqrt{17}$$