Solve (x-frac{3-sqrt{5}}{2})(x-frac{sqrt{5}+3}{2}

$$( x - \frac { 3 - \sqrt { 5 } } { 2 } ) ( x - \frac { \sqrt { 5 } + 3 } { 2 }$$

Evaluate

$$x^{2}-3x+1$$

Short Solution Steps

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

$$\left(\frac{2x}{2}-\frac{3-\sqrt{5}}{2}\right)\left(x-\frac{\sqrt{5}+3}{2}\right)$$

Since \frac{2x}{2} and \frac{3-\sqrt{5}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x-\left(3-\sqrt{5}\right)}{2}\left(x-\frac{\sqrt{5}+3}{2}\right)$$

Do the multiplications in 2x-\left(3-\sqrt{5}\right).

$$\frac{2x-3+\sqrt{5}}{2}\left(x-\frac{\sqrt{5}+3}{2}\right)$$

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

$$\frac{2x-3+\sqrt{5}}{2}\left(\frac{2x}{2}-\frac{\sqrt{5}+3}{2}\right)$$

Since \frac{2x}{2} and \frac{\sqrt{5}+3}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x-3+\sqrt{5}}{2}\times \frac{2x-\left(\sqrt{5}+3\right)}{2}$$

Do the multiplications in 2x-\left(\sqrt{5}+3\right).

$$\frac{2x-3+\sqrt{5}}{2}\times \frac{2x-\sqrt{5}-3}{2}$$

Multiply \frac{2x-3+\sqrt{5}}{2} times \frac{2x-\sqrt{5}-3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2x-3+\sqrt{5}\right)\left(2x-\sqrt{5}-3\right)}{2\times 2}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(2x-3+\sqrt{5}\right)\left(2x-\sqrt{5}-3\right)}{4}$$

Apply the distributive property by multiplying each term of 2x-3+\sqrt{5} by each term of 2x-\sqrt{5}-3.

$$\frac{4x^{2}-2x\sqrt{5}-6x-6x+3\sqrt{5}+9+2\sqrt{5}x-\left(\sqrt{5}\right)^{2}-3\sqrt{5}}{4}$$

Combine -6x and -6x to get -12x.

$$\frac{4x^{2}-2x\sqrt{5}-12x+3\sqrt{5}+9+2\sqrt{5}x-\left(\sqrt{5}\right)^{2}-3\sqrt{5}}{4}$$

Combine -2x\sqrt{5} and 2\sqrt{5}x to get 0.

$$\frac{4x^{2}-12x+3\sqrt{5}+9-\left(\sqrt{5}\right)^{2}-3\sqrt{5}}{4}$$

The square of \sqrt{5} is 5.

$$\frac{4x^{2}-12x+3\sqrt{5}+9-5-3\sqrt{5}}{4}$$

Subtract 5 from 9 to get 4.

$$\frac{4x^{2}-12x+3\sqrt{5}+4-3\sqrt{5}}{4}$$

Combine 3\sqrt{5} and -3\sqrt{5} to get 0.

$$\frac{4x^{2}-12x+4}{4}$$

Divide each term of 4x^{2}-12x+4 by 4 to get 1-3x+x^{2}.

$$1-3x+x^{2}$$

Factor

$$\left(x-\frac{3-\sqrt{5}}{2}\right)\left(x-\frac{\sqrt{5}+3}{2}\right)$$