Intersection relations of quadratic functions X1 + X2, x1 × x2

Intersection relations of quadratic functions X1 + X2, x1 × x2

f(x)=ax^2+bx+c(a≠0)
Then: X1 + x2 = - B / A
x1*x2=c/a

Given the quadratic function y = x ^ 2-mx + m-2, if there are two intersections (x1,0), (x2,0) between the function image and x-axis, use m to represent X1 ^ 2 + x2 ^ 2 and find its minimum value
It's a big question. We need to answer it in detail

According to Weida's theorem: X1 + x2 = m / 2, x1x2 = m-2
x1^2+x2^2=(x1+x2)^2-2x1x2=m^2/4-2m+4
Minimum value: w = (4ac-b ^ 2) / 4A = 0

Find the minimum distance between the image of the quadratic function y = x ^ 2 + 3 (m-2) x + m (M-4) and the intersection of the X axis. At this time, the value of M is?

X ^ 2 + 3 (m-2) x + m (M-4) = 0
x1+x2=3（2-m）,x1x2=m（m-4）,
Then the square of the distance between two intersections (x1-x2) ^ 2 = (x1 + x2) ^ 2-4x1x2 = 9 (m-2) ^ 2-4m (M-4)
=5m^2+20m+36
=5(m+2)^2+16,
So when m = - 2, the minimum distance is 4