If the roots of the equation x3+bx2+cx+d=0 are in arithmetic progression, then b, c and d satisfy the relation

Question

If the roots of the equation x3+bx2+cx+d=0 are in arithmetic progression, then b, c and d satisfy the relation (A) 2b2-27d=9bc (B) 2b3-27d=9bc (C) 2b2+27d=9bc (D) 2b3+27d=9bc

Tardigrade Admin · Accepted Answer

Let α ,β ,&gamma; be the required roots α +β +&gamma; =-b, α β +β &gamma; +&gamma; α =c , α β &gamma; =-d Also, α +&gamma; =2β ⇒ 3β =-b β (.α +&gamma; .)+&gamma; α =c⇒ ((- b/3))2((- b/3))+&gamma; α =c ⇒ (2 b2/9)-(d/β )=c⇒ (2 b2/9)-(d/- (b)3)=c ⇒ (2 b2/9)+(3 d/b)=c⇒ 2b3+27d=9bc

Q. If the roots of the equation $x^{3} + b x^{2} + c x + d = 0$ are in arithmetic progression, then $b, c$ and $d$ satisfy the relation

$2 b^{2} - 27 d = 9 b c$

$2 b^{3} - 27 d = 9 b c$

$2 b^{2} + 27 d = 9 b c$

$2 b^{3} + 27 d = 9 b c$

Q. If the roots of the equation x3+bx2+cx+d=0 are in arithmetic progression, then b,c and d satisfy the relation

2b2−27d=9bc

2b3−27d=9bc

2b2+27d=9bc

2b3+27d=9bc

Let α,β,γ be the required roots α+β+γ=−b,αβ+βγ+γα=c , αβγ=−d Also, α+γ=2β⇒3β=−b β(α+γ)+γα=c⇒(3−b​)2(3−b​)+γα=c ⇒92b2​−βd​=c⇒92b2​−−3b​d​=c ⇒92b2​+b3d​=c⇒2b3+27d=9bc

Q. If the roots of the equation $x^{3} + b x^{2} + c x + d = 0$ are in arithmetic progression, then $b, c$ and $d$ satisfy the relation

$2 b^{2} - 27 d = 9 b c$

$2 b^{3} - 27 d = 9 b c$

$2 b^{2} + 27 d = 9 b c$

$2 b^{3} + 27 d = 9 b c$