# How do you find the discriminant of 7x^2+6x+2=0 and use it to determine if the equation has one, two real or two imaginary roots?

Mar 18, 2017

$7 {x}^{2} + 6 x + 2$ has complex number roots

#### Explanation:

Consider the standard format of $y = a {x}^{2} + b x + c$

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

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

If this is 0 then the vertex is actually on the on the axis so has 1 point

If this is negative then the roots are complex numbers

If this is positive and greater than 0 then it has two roots
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Added comment: I have had people argue that when the discriminant is 0 there is still 2 roots but they are the same value. This condition is called duplicity.
I do not like this!
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So for this equation:

Discriminant $\to {b}^{2} - 4 a c = {\left(6\right)}^{2} - 4 \left(7\right) \left(2\right) = 36 - 56 < 0$

So $7 {x}^{2} + 6 x + 2$ has complex number roots

Mar 18, 2017

The solutions are $S = \left\{- \frac{3}{7} + \frac{\sqrt{5}}{7} i , - \frac{3}{7} - \frac{\sqrt{5}}{7} i\right\}$

#### Explanation:

$a {x}^{2} + b x + c = 0$

Here, we have

$7 {x}^{2} + 6 x + 2 = 0$

The discriminant is

$\Delta = {b}^{2} - 4 a c = 36 - 4 \cdot 7 \cdot 2 = 36 - 56 = - 20$

As $\Delta < 0$, there are no real roots.

The roots are imaginary

$x = \frac{- b \pm \sqrt{\Delta}}{2 a}$

$x = \frac{- 6 \pm i 2 \sqrt{5}}{14}$

${x}_{1} = - \frac{3}{7} + \frac{\sqrt{5}}{7} i$

${x}_{2} = - \frac{3}{7} - \frac{\sqrt{5}}{7} i$