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

`color(blue)("Determine the discriminant")`

Taken as: `8n^2-6n+3=5n^2`

Subtract `5n^2` from both sides

`color(red)(3n^2-6n+3=0) larr" Use this one"`

Compare to `y=ax^2+bx+c = 0`

Where `x=(-b+-sqrt(b^2-4ac))/(2a)`

In your case `x` is `n`

The discriminant part is the `b^2-4ac` giving

`(-6)^2-4(3)(3) = 36-36=0`

So the discriminant is 0
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`color(blue)("Further comments")`

`color(red)("IF")` the graph crosses the x-axis in two different places the quadratic formula results in a solution of format:

`"some value " +-" some other value"`

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

`"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'.