# Homework Help: Cubic equation roots.

1. Mar 7, 2010

### ibysaiyan

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. The attempt at a solution

2. Mar 7, 2010

### 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.

3. Mar 7, 2010

### ibysaiyan

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 :(.

4. Mar 7, 2010

### rock.freak667

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?

5. Mar 7, 2010

3a?

6. Mar 7, 2010

### rock.freak667

Right yes good 3α.

So ∑α=3α=3, what is α then?

7. Mar 8, 2010

### ibysaiyan

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 =/ .

8. Mar 8, 2010

### 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.

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