Solve y=(x-frac{9-sqrt{5}}{8})(x-frac{9+sqrt{5}}{8})

$$y = ( x - \frac { 9 - \sqrt { 5 } } { 8 } ) ( x - \frac { 9 + \sqrt { 5 } } { 8 } )$$

Solve for y

$$y=x^{2}-\frac{9x}{4}+\frac{19}{16}$$

Short Solution Steps

Use the distributive property to multiply x-\frac{9-\sqrt{5}}{8} by x-\frac{9+\sqrt{5}}{8}.

$$y=x^{2}+x\left(-\frac{9+\sqrt{5}}{8}\right)+\left(-\frac{9-\sqrt{5}}{8}\right)x+\left(-\frac{9-\sqrt{5}}{8}\right)\left(-\frac{9+\sqrt{5}}{8}\right)$$

Express x\left(-\frac{9+\sqrt{5}}{8}\right) as a single fraction.

$$y=x^{2}+\frac{-x\left(9+\sqrt{5}\right)}{8}+\left(-\frac{9-\sqrt{5}}{8}\right)x+\left(-\frac{9-\sqrt{5}}{8}\right)\left(-\frac{9+\sqrt{5}}{8}\right)$$

Express \left(-\frac{9-\sqrt{5}}{8}\right)x as a single fraction.

$$y=x^{2}+\frac{-x\left(9+\sqrt{5}\right)}{8}+\frac{-\left(9-\sqrt{5}\right)x}{8}+\left(-\frac{9-\sqrt{5}}{8}\right)\left(-\frac{9+\sqrt{5}}{8}\right)$$

Multiply -\frac{9-\sqrt{5}}{8} times -\frac{9+\sqrt{5}}{8} by multiplying numerator times numerator and denominator times denominator.

$$y=x^{2}+\frac{-x\left(9+\sqrt{5}\right)}{8}+\frac{-\left(9-\sqrt{5}\right)x}{8}+\frac{\left(9-\sqrt{5}\right)\left(9+\sqrt{5}\right)}{8\times 8}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{2} times \frac{8\times 8}{8\times 8}.

$$y=\frac{x^{2}\times 8\times 8}{8\times 8}+\frac{-x\left(9+\sqrt{5}\right)}{8}+\frac{-\left(9-\sqrt{5}\right)x}{8}+\frac{\left(9-\sqrt{5}\right)\left(9+\sqrt{5}\right)}{8\times 8}$$

Since \frac{x^{2}\times 8\times 8}{8\times 8} and \frac{\left(9-\sqrt{5}\right)\left(9+\sqrt{5}\right)}{8\times 8} have the same denominator, add them by adding their numerators.

$$y=\frac{x^{2}\times 8\times 8+\left(9-\sqrt{5}\right)\left(9+\sqrt{5}\right)}{8\times 8}+\frac{-x\left(9+\sqrt{5}\right)}{8}+\frac{-\left(9-\sqrt{5}\right)x}{8}$$

Do the multiplications in x^{2}\times 8\times 8+\left(9-\sqrt{5}\right)\left(9+\sqrt{5}\right).

$$y=\frac{64x^{2}+81+9\sqrt{5}-9\sqrt{5}-5}{8\times 8}+\frac{-x\left(9+\sqrt{5}\right)}{8}+\frac{-\left(9-\sqrt{5}\right)x}{8}$$

Combine like terms in 64x^{2}+81+9\sqrt{5}-9\sqrt{5}-5.

$$y=\frac{64x^{2}+76}{8\times 8}+\frac{-x\left(9+\sqrt{5}\right)}{8}+\frac{-\left(9-\sqrt{5}\right)x}{8}$$

Factor the expressions that are not already factored in \frac{64x^{2}+76}{8\times 8}.

$$y=\frac{4\left(16x^{2}+19\right)}{8\times 8}+\frac{-x\left(9+\sqrt{5}\right)}{8}+\frac{-\left(9-\sqrt{5}\right)x}{8}$$

Cancel out 4 in both numerator and denominator.

$$y=\frac{16x^{2}+19}{2\times 8}+\frac{-x\left(9+\sqrt{5}\right)}{8}+\frac{-\left(9-\sqrt{5}\right)x}{8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\times 8 and 8 is 2\times 8. Multiply \frac{-x\left(9+\sqrt{5}\right)}{8} times \frac{2}{2}.

$$y=\frac{16x^{2}+19}{2\times 8}+\frac{2\left(-1\right)x\left(9+\sqrt{5}\right)}{2\times 8}+\frac{-\left(9-\sqrt{5}\right)x}{8}$$

Since \frac{16x^{2}+19}{2\times 8} and \frac{2\left(-1\right)x\left(9+\sqrt{5}\right)}{2\times 8} have the same denominator, add them by adding their numerators.

$$y=\frac{16x^{2}+19+2\left(-1\right)x\left(9+\sqrt{5}\right)}{2\times 8}+\frac{-\left(9-\sqrt{5}\right)x}{8}$$

Do the multiplications in 16x^{2}+19+2\left(-1\right)x\left(9+\sqrt{5}\right).

$$y=\frac{16x^{2}+19-18x-2x\sqrt{5}}{2\times 8}+\frac{-\left(9-\sqrt{5}\right)x}{8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\times 8 and 8 is 2\times 8. Multiply \frac{-\left(9-\sqrt{5}\right)x}{8} times \frac{2}{2}.

$$y=\frac{16x^{2}+19-18x-2x\sqrt{5}}{2\times 8}+\frac{2\left(-1\right)\left(9-\sqrt{5}\right)x}{2\times 8}$$

Since \frac{16x^{2}+19-18x-2x\sqrt{5}}{2\times 8} and \frac{2\left(-1\right)\left(9-\sqrt{5}\right)x}{2\times 8} have the same denominator, add them by adding their numerators.

$$y=\frac{16x^{2}+19-18x-2x\sqrt{5}+2\left(-1\right)\left(9-\sqrt{5}\right)x}{2\times 8}$$

Do the multiplications in 16x^{2}+19-18x-2x\sqrt{5}+2\left(-1\right)\left(9-\sqrt{5}\right)x.

$$y=\frac{16x^{2}+19-18x-2x\sqrt{5}-18x+2\sqrt{5}x}{2\times 8}$$

Combine like terms in 16x^{2}+19-18x-2x\sqrt{5}-18x+2\sqrt{5}x.

$$y=\frac{16x^{2}+19-36x}{2\times 8}$$

Factor the expressions that are not already factored in \frac{16x^{2}+19-36x}{2\times 8}.

$$y=\frac{16\left(x-\left(-\frac{1}{8}\sqrt{5}+\frac{9}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{5}+\frac{9}{8}\right)\right)}{2\times 8}$$

Cancel out 2\times 8 in both numerator and denominator.

$$y=\left(x-\left(-\frac{1}{8}\sqrt{5}+\frac{9}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{5}+\frac{9}{8}\right)\right)$$

Expand the expression.

$$y=x^{2}-\frac{9}{4}x+\frac{19}{16}$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{64y+5}+9}{8}$$

$$x=\frac{\sqrt{64y+5}+9}{8}$$

Solve for x

$$x=\frac{-\sqrt{64y+5}+9}{8}$$

$$x=\frac{\sqrt{64y+5}+9}{8}\text{, }y\geq -\frac{5}{64}$$