### Instruction: Sviatoslav solved the quadratic equation $x^2-x-1=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+a)^2 = b,$$where $a$ and $b$ are constants.

What is $b$?### Response: I want to find out how Sviatoslav completed the square for the equation $x^2-x-1=0$.
To do that, I need to move the constant term $-1$ to the right side and then add the same value to both sides to make the left side a perfect square.
The value I need to add is half of the coefficient of $x$, which is $-1$, squared. That is, $\left(-\frac{1}{2}\right)^2 = \frac{1}{4}$.
So I get the equation $$x^2-x + \frac{1}{4} = 1 + \frac{1}{4},$$ which simplifies to $$\left(x-\frac{1}{2}\right)^2 = \frac{5}{4}.$$
Comparing this equation with $(x+a)^2 = b$, I can see that $a = -\frac{1}{2}$ and $b = \frac{5}{4}$.