Find Real Values of a for 3 Distinct Roots of x^2-3x+a=0

[utkarshakash]

Homework Statement

The real values of a for which the equation x^2-3x+a=0 has three real and distinct roots is

The Attempt at a Solution

I started by writing the sign scheme of f'(x). But it is of no help to me. It will merely tell me the intervals in which f(x) increases or decreases. Also, if there are three distinct roots of f(x) then there must be two extrema.

 

[CAF123]

Did you write down the problem statement correctly? If a is a real constant, then the equation x² -3x + a = 0 gives the roots of a quadratic, which by the FTOA, there is exactly 2 roots in $\mathbb{C}$. Did you mean to write 'two real and distinct roots?

 

[utkarshakash]

Did you write down the problem statement correctly? If a is a real constant, then the equation x² -3x + a = 0 gives the roots of a quadratic, which by the FTOA, there is exactly 2 roots in $\mathbb{C}$. Did you mean to write 'two real and distinct roots?

I'm really sorry. It is x^3.

 

[Borek]

Sketch the plot - for three distinct roots the extremes must be on both sides of the abscissa. And "a" changes position of the plot with regard to abscissa, so for some values you will have just one root, for some values two roots, and for some values three roots. Think how these things change depending on the number of extreme values and how the number of extreme values depend on "a".

 

[CAF123]

You could try finding what values of a are roots of that function. That will allow you to factor out an (x±a) and you are automatically left with a simple quadratic