What is minimum positive value of p so that the quadratic polynomial
\[ \dfrac34 x^2 - (2p+1)x + p^2 +2 = 0 \]
has real roots?
+128475 To have real roots, the discriminant must be >= to 0
Therefore
(2p+1)^2 - 4 ( 3/4) (p^2 + 2) >= 0
4p^2 + 4p + 1 - 3 (p^2 + 2) > = 0
4p^2 + 4p + 1 - 3p^2 - 6 >= 0
p^2 + 4p - 5 >= 0 factor
(p + 5) ( p -1) > = 0
Note that this will be true when p = [ 1, inf) or when p = ( - inf, -5 ]
p = 1 is the minimum positive value that will produce real roots
