help finding the directrix, focus, and roots of a parabola?

Question

find (a) the directrix, (b) the focus, and (c) the roots of the parabola y = x^2 - 5x + 5

SORRY ITS +4 INSTEAD OF 5 SO INSTEAD IT IS

Y = X^2 - 5X + 4

Answer 1

Given the parabola y = x² - 5x + 5

Complete the square.

y = x² - 5x + 5
y - 5 + 25/4 = (x² - 5x + 25/4)
y + 5/4 = (x - 5/2)²

The vertex (h,k) = (5/2, -5/4).

The vertex form of a parabola with vertex (h,k) is

4p(y - k) = (x - h)²

For the given parabola we have

y + 5/4 = (x - 5/2)²

4p = 1
p = 1/4

The directed distance from the vertex to the focus is p.

The focus is (h, k+p) = (5/2, -5/4 + 1/4) = (5/2, -1).

The directed distance from the vertex to the directrix is -p.

The directrix is y = k - p = -5/4 - 1/4
y = -3/2

For the zeros of the parabola set it equal to zero.

y = x² - 5x + 5
0 = x² - 5x + 5

x = [5 ± √(25 - 4*1*5)] / 2 = (5 ± √5)/2

Answer 2

x^2 - 5x + 5
= x^2 - 5x + 6.25 - 1.25
= (x - 2.5)^2 - 1.25
Therefore, the vertex is at (2.5, -1.25)
4P = 1
P = 0.25
a) The directrix: y = -1.5
b) Focus: (2.5, -1)
c) The roots: x = 2.5±√1.25