X1 and X2 are the two real roots of the equation x ^ 2 - (2m-1) x + (m ^ 2 + 2m-4) = 0, and find the minimum value of X1 ^ 2 + x2 ^ 2

X1 and X2 are the two real roots of the equation x ^ 2 - (2m-1) x + (m ^ 2 + 2m-4) = 0, and find the minimum value of X1 ^ 2 + x2 ^ 2

X1, X2 are the two real roots of the equation x & # 178; - (2m-1) x + (M & # 178; + 2m-4) = 0, so X1 + x2 = 2m-1, x1x2 = M & # 178; + 2m-4
Δ=(2m-1)²-4(m²+2m-4)=17-12m≥0
So m ≤ 17 / 12
So X1 & # 178; + x2 & # 178; = (x1 + x2) &# 178; - 2x1x2 = (2m-1) &# 178; - 2 (M & # 178; + 2m-4) = 2m & # 178; - 8m + 9
=2(m-2)²+1
Because m ≤ 17 / 12
So the minimum value of X1 & # 178; + x2 & # 178; is 2 (17 / 12-2) &# 178; + 1 = 121 / 72