Answer to Question #240268 in Algebra for Ahmed

Question #240268

Equation X^3 + 21X^2 + p = 0$$ has three different real solutions that form an arithmetic progression. Find the largest value of parameter p.


1
Expert's answer
2021-09-22T05:21:43-0400

Solution:

x3+21x2+p=0x^3+21x^2+p=0

It can be written as (xa)(xb)(xc)=0(x-a)(x-b)(x-c)=0

Here, roots are a,b,ca,b,c.

Then, x3+21x2+p=(xa)(xb)(xc)x^3+21x^2+p=(x-a)(x-b)(x-c)

x3+21x2+p=(xa)(x2x(b+c)+bc)x3+21x2+p=x3ax2x2(b+c)+ax(b+c)+xbcabcx3+21x2+p=x3+x2(abc)+x(ab+ac+bc)abc\Rightarrow x^3+21x^2+p=(x-a)(x^2-x(b+c)+bc) \\ \Rightarrow x^3+21x^2+p=x^3-ax^2-x^2(b+c)+ax(b+c)+xbc-abc \\ \Rightarrow x^3+21x^2+p=x^3+x^2(-a-b-c)+x(ab+ac+bc)-abc

On comparing both sides,

abc=21; ab+ac+bc=0;abc=pa+b+c=21 ...(i); ab+ac+bc=0;abc=p-a-b-c=21;\ ab+ac+bc=0;-abc=p \\\Rightarrow a+b+c=-21\ ...(i);\ ab+ac+bc=0;abc=-p

It is given that the roots are in AP.

a+c=2b ...(ii)a+c=2b\ ...(ii)

using (ii) in (i),

3b=21b=73b=-21 \\ \Rightarrow b=-7

Put this value in (ii),

a+c=14a+c=-14

Also, abc=pabc=-p

ac(7)=p\Rightarrow ac(-7)=-p [Using b=-7]

ac=p7\Rightarrow ac=\dfrac p7

Also, ab+ac+bc=0ab+ac+bc=0

7a+ac7c=0\Rightarrow -7a+ac-7c=0 [Using b=-7]

7(a+c)=ac7(14)=p7p=686\Rightarrow 7(a+c)=ac \\\Rightarrow 7(-14)=\dfrac p7 \\ \Rightarrow p=-686


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