# What is the discriminent and the solutions of 2x^2+3x+5?

Mar 7, 2018

$x = - \frac{3}{4} \pm \frac{\sqrt{31}}{4} i$

#### Explanation:

$\textcolor{b l u e}{\text{Determining the discriminant}}$

Consider the structure $y = a {x}^{2} + b x + c$

where $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

The discriminant is the part ${b}^{2} - 4 a c$

So in this case we have:

a=2; b=3 and c=5

Thus the discriminant part ${b}^{2} - 4 a c \to {\left(3\right)}^{2} - 4 \left(2\right) \left(5\right) = - 31$

As this is negative it means that the solution to $a {x}^{2} + b x + c$ is such that $x$ is not in the set of Real Numbers but is in the set of Complex numbers.
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$\textcolor{b l u e}{\text{Determine the solution for } a {x}^{2} + b x + c = 0}$

#Using the above formula we have:

$x = \frac{- 3 \pm \sqrt{- 31}}{4}$

$x = - \frac{3}{4} \pm \frac{\sqrt{31 \times \left(- 1\right)}}{4}$

$x = - \frac{3}{4} \pm \frac{\sqrt{31}}{4} i$