Cubic Equation Roots: Solving for α and β | p Value Calculation

[ibysaiyan]

Homework Statement

The equation
x3 − 3x2 + px + 4 = 0,
where p is a constant, has roots α −β , α and α + β , where β > 0.
(a) Find the values of α and β .
(b) Find the value of p.
how do i start off? all i know is that sigma a= -b/a and ab= c/a and ab(gamma) = -d/a .
Would this be one of the ways to do it: [x-(a-β)] [x-(a+β)] (x-a).
Thanks.

Homework Equations

The Attempt at a Solution

 

[rock.freak667]

You know that ∑α=-b/a, so use that to find α. ∑αβ=c/a and ∑αβγ=-d/a.

So start off using the first method. Sum of all of the roots and equate it to 3. Do a similar exercise for the product of the roots.

 

[ibysaiyan]

You know that ∑α=-b/a, so use that to find α. ∑αβ=c/a and ∑αβγ=-d/a.

So start off using the first method. Sum of all of the roots and equate it to 3. Do a similar exercise for the product of the roots.

Oo is this how it works: ∑α=-b/a => -(-3)/1 = 3.
to find b: ∑ab=c/a
= ∑a.∑b= c/a.
3∑b=px ... b=px/3. <--- sorry for such a weird presentation. Not sure :(.

 

[rock.freak667]

Oo is this how it works: ∑α=-b/a => -(-3)/1 = 3.
to find b: ∑ab=c/a
= ∑a.∑b= c/a.
3∑b=px ... b=px/3. <--- sorry for such a weird presentation. Not sure :(.

Yes, but ∑α is the sum of the roots, your roots are α−β , α and α + β, what are the sum of the roots?

 

[ibysaiyan]

Yes, but ∑α is the sum of the roots, your roots are α−β , α and α + β, what are the sum of the roots?

3a?

 

[rock.freak667]

3a?

Right yes good 3α.

So ∑α=3α=3, what is α then?

 

[ibysaiyan]

Right yes good 3α.

So ∑α=3α=3, what is α then?

Oh! its one.. so to find out constant b would i mutiply out my roots:when a=1
(1-b) (1+b) (1) = i think its wrong =/ .

 

[HallsofIvy]

Another way to do this would be to set
$$(x- \alpha- \beta)(x- \alpha)(x- \alpha+ \beta)= x^2- 3x^2+ px+ 4$$
multiply it out and set corresponding coefficients equal. That gives you three equations for $\alpha$, $\beta$, and p.