Solving for Roots of $y^2 - \sqrt{5} y - y + \sqrt{5} = 0$

To find the roots of the equation $y^2 - \sqrt{5} y - y + \sqrt{5} = 0$, we can use factoring by grouping.

First, group the terms of the equation:

$(y^2 - \sqrt{5} y) + (-y + \sqrt{5}) = 0$

Next, factor out the greatest common divisor from each group:

$y(y - \sqrt{5}) - 1(y - \sqrt{5}) = 0$

Notice that $(y - \sqrt{5})$ is a common binomial factor. Factor it out:

$(y - \sqrt{5})(y - 1) = 0$

For the product of two factors to equal zero, at least one of the factors must be zero. Set each factor equal to zero and solve for $y$:

• Setting the first factor to zero: $y - \sqrt{5} = 0 \implies y = \sqrt{5}$
• Setting the second factor to zero: $y - 1 = 0 \implies y = 1$

The roots of the equation are $\sqrt{5}$ and $1$.