What is minimum positive value of p so that the quadratic polynomial
\[\dfrac34 x^2 - (2p+2)x + p^2 +2 = 0\]
has real roots?
+208 Try and use the quadratic formula. Remember that if: \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) has real roots, which part of the formula must be positive? Then substitute all the terms in.
+128460 To have real roots, the discriminant must be ≥ 0
So
(2p+2)^2 - 4 (3/4) ( p^2 + 2) ≥ 0
4p^2 + 8p + 4 - 3p^2 - 6 ≥ 0
p^2 + 8p - 2 ≥ 0 complete the square on p
p^2 + 8p + 16 ≥ 2 + 16
(p + 4)^2 ≥ 18 take the positive root
p + 4 ≥ sqrt (18)
p ≥ sqrt (18) - 4 ≈ .2426 = the smallest approx. + value of p that will give real roots
