a quadratic equation for two unknown number

Question

find values of $p$ such that the equation $4x^2 + 3px - 2p = 0$ has? below are a few choices of the value p:

a) 2 real roots

b) 1 real roots

c) no roots or complex roots

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so far i did for a) 2 real roots

$$(3p)^2 - 4(4)(-2p) > 0\\ 9p^2 +32p > 0\\ 9\left( p^2+\frac{32p}{9} \right) > 0\\ p^2+\frac{32p}{9} > 0\\ p^2 +\frac{32p}{9} + \left(\frac{32}{18}\right)^2 > 0\\ \left(p + \frac{32}{18} \right) ^2 > 0 \\ p + \frac{32}{18} > 0\\ p > -\frac{32}{18} $$

Answer 1

The roots are given by the quadratic formula: $$ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\ = \frac{-3C \pm \sqrt{9C^2 + 32p}}{2a} $$ If the thing under the square root is positive, you'll get two roots. Under what conditions will you get just one? Under what condition will you get none?

Answer 2

$D=(3p)^2+4\cdot4\cdot2p=9p^2+32p=p(9p+32)$

if $D>0$ then there is 2(different) real roots

$D=9p^2+32p=p(9p+32)>0$

$p\in(-\infty;-32/9)\cup(0;+\infty)$

if $D=0$ then there is 1(or 2 equal) real root

$p=0$ and $p=-32/9$

if $D<0$ then there is no roots or 2 complex roots

$p\in(-32/9;0)$