Submitted by Benjamin 1 on Thu, 08/31/2017 - 19:43

For assignment 9, although I see where all the values come from, I am struggling to see how (27b^2) + (4a^3b) < 0 implies that there are 3 distinct real roots.

If I use the cubic x^3 - 4x + 4 then

(27b^2)+(4a^3)(b) = -592, which satisfies the above inequality, yet there exists only one distinct real root?

On a side note, I am working through the STEP I assignments in chronological order and am at Assignment 9. At this stage I can complete about half to 2/3 of the question - is this a good/bad sign? Is it bad that I cannot complete the whole question at this stage?

Ben

## hope this helps

The step question only talks about functions with $x^3+ax^2+b$, your example doesn't have an $x^2$ value.

The number of real roots of a cubic can be found by looking at the turning points of the function. If for example they are both above or below one axis it is clear to see that the cubic has only one solution. Only if one is above and one below, does the cubic have three roots.

When differentiated the TP are $(0,b)$ and $(-2a/3, 4a^3/3 +b)$

look at the cases when b is positive and negative, and whether the y-value of the second TP would need to be positive or negative. You will get two different inequalities, which can be simplified into one.

## Great, thanks!

Great, thanks!

## Minor quibble with "The step

Minor quibble with "The step question only talks about functions with $x^3+ax^2+bx$, your example doesn't have an $x^2$ value" - the OP's example does have an $x^2$ 'value', it's $a=0$, the difference is that he has a non-zero $x$ coefficient which is where the example fails to emulate the STEP question.