# How do you use the discriminant to determine the nature of the solutions given  –3p^2 – p + 2 = 0?

Aug 26, 2016

As the discriminant is positive the nature of the solutions are such that the graph crosses the x-axis so $x \in \mathbb{R}$ for $- 3 {p}^{2} - p + 2 = 0$

#### Explanation:

Standard for equation but with $p$ instead of $x$:$\to a {p}^{2} - p + 2 = 0$

where a = -3; b= -1; c=+2

$\implies p = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

Thus the discriminant ${b}^{2} - 4 a c \text{ "->" } {\left(- 3\right)}^{2} - 4 \left(- 3\right) \left(+ 2\right) = + 33$

The nature if the solution is that the plot does cross the x-axis. So there are values of $x$ where $- 3 {p}^{2} - p + 2 = 0$ is true

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By the way. If the coefficient of ${x}^{2}$ is positive then the graph is of general shape $\cup$.

However, the coefficient is -3 thus negative. So the graph is of general shape $\cap$