Question

if alpha beta and gamma are the roots of the equation X cube minus 3 X square + X + 5 equal to zero then y is equals to sigma of alpha square plus alpha beta gamma satisfying the equation

Answer 1

Step-by-step explanation:

α,β and γ are the roots of the equation x3+3ax2+3bx+c=0.

⇒α+β+γ=−3a,αβ+βγ+αγ=3b, and αβγ=−c.

It is given that α,β and γ are in HP.

⇒1α,1β and 1γ are in AP.

⇒1α+1γ=2β⇒α+γαγ=2β

⇒αβ+βγ=2αγ⇒αβ+βγ+αγ=3αγ=3αβγβ.

⇒3b=−3cβ⇒β=−cb.

Edit: Amitabha Tripathi has asked “What if b=0?"

We have already shown that if α,β and γ are in HP, then 3b=−3cβ.

Hence, b=0⇒3cβ=0.

⇒c=0 and β≠0.

⇒ The given cubic equation then reduces to x3+3ax2=0.

⇒x2(x+3a)=0.

⇒ The roots are 0,0 and −3a.

⇒ The reciprocals of two of the roots are not defined.

⇒ The roots cannot be in HP.

However, it is given that the roots are in HP.

⇒b cannot be equal to 0 if the roots are to be in HP.

Hope it will help you

Answer 2

Step-by-step explanation:

α,β and γ are the roots of the equation x3+3ax2+3bx+c=0.

⇒α+β+γ=−3a,αβ+βγ+αγ=3b, and αβγ=−c.

It is given that α,β and γ are in HP.

⇒1α,1β and 1γ are in AP.

⇒1α+1γ=2β⇒α+γαγ=2β

⇒αβ+βγ=2αγ⇒αβ+βγ+αγ=3αγ=3αβγβ.

⇒3b=−3cβ⇒β=−cb.

Hope it will help you