# How do you find the value of the discriminant and state the type of solutions given 8b^2-6n+3=5b^2?

Jan 31, 2017

Assumption: The only variable is $n$

#### Explanation:

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

Taken as: $8 {n}^{2} - 6 n + 3 = 5 {n}^{2}$

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

$\textcolor{red}{3 {n}^{2} - 6 n + 3 = 0} \leftarrow \text{ Use this one}$

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

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

In your case $x$ is $n$

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

${\left(- 6\right)}^{2} - 4 \left(3\right) \left(3\right) = 36 - 36 = 0$

So the discriminant is 0
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$\textcolor{b l u e}{\text{Further comments}}$

$\textcolor{red}{\text{IF}}$ the graph crosses the x-axis in two different places the quadratic formula results in a solution of format:

$\text{some value " +-" some other value}$

However, if the discriminant is 0 as in this case you have:

$\text{some value " +-" } 0$

giving a single value solution. Consequently the x-axis behave like a tangent to the curve at the min/max point.

So the curve does not actually cross the axis but it more like the two coincide.

However; some people argue that their is always 2 solution but the condition we have hear has the sate of 'duality'.

I am guessing this means that the two solutions happen to coincide thus look as though there is one. This is higher maths 'stuff'.