How do you find the value of the discriminant and determine the nature of the roots 8b^2-6b+3=5b^2?

Sep 19, 2016

The nature of the 'roots' is that 'they' both have the same value. Some people refer to this as duplicity.

In effect, there is only one root.

Explanation:

Given:$\text{ } 8 {b}^{2} - 6 b + 3 = 5 {b}^{2}$

Subtract $5 {b}^{2}$ from both sides

$3 {b}^{2} - 6 b + 3 = 0$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

To avoid confusion let $b = t$

Then we have: $3 {t}^{2} - 6 t + 3$

Compare to $y = a {t}^{2} + b t + c = 0$

$a = 3 \text{; "b=-6"; } c = 3$

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

$\implies t = \frac{+ 6 \pm \sqrt{{6}^{2} - 4 \left(3\right) \left(3\right)}}{2 \left(3\right)}$

The discriminant is:

${b}^{2} - 4 a c \to 36 - 36 = 0$

When the discriminant is 0 it means that the x-axis is tangential to the curve at the maximum/minimum point.

The there is a single value solution.