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Question Number 95373 by Don08q last updated on 24/May/20

Find the value of m for which the roots of the equation x^3 + 6x^2 + 11x +m = 0 form a linear sequence.

$$\: \\ $$$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{m}\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{roots} \\ $$$$\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:{x}^{\mathrm{3}} \:+\:\mathrm{6}{x}^{\mathrm{2}} \:+\:\mathrm{11}{x}\:+{m}\:=\:\mathrm{0} \\ $$$$\:\mathrm{form}\:\mathrm{a}\:\mathrm{linear}\:\mathrm{sequence}. \\ $$$$ \\ $$

Commented by Rasheed.Sindhi last updated on 24/May/20

Linear sequence means AP?

$$\mathrm{Linear}\:\mathrm{sequence}\:\mathrm{means}\:\mathrm{AP}? \\ $$

Commented by Don08q last updated on 24/May/20

Yes please

$$\mathrm{Yes}\:\mathrm{please} \\ $$

Answered by prakash jain last updated on 24/May/20

Let roots be a−d,a,a+d sum of roots=−(coefficient of x^2 ) a−d+a+a+d=−6⇒a=−2 sum of product of roots taken2 at time =+(coefficient of x) (a−d)(a)+a(a+d)+(a−d)(a+d)=11 2a^2 +a^2 −d^2 =11 3a^2 −d^2 =11 d^2 =3×4−11=1 product of roots=−(constant term) m=−(a−d)(a)(a+d) =−(a^2 −d^2 )a =−(4−1)×6

$$\mathrm{Let}\:\mathrm{roots}\:\mathrm{be}\:{a}−{d},{a},{a}+{d} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{roots}=−\left(\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{2}} \right) \\ $$$${a}−{d}+{a}+{a}+{d}=−\mathrm{6}\Rightarrow{a}=−\mathrm{2} \\ $$$$\mathrm{sum}\:\mathrm{of}\:\mathrm{product}\:\mathrm{of}\:\mathrm{roots}\:\mathrm{taken2}\:\mathrm{at}\:\mathrm{time} \\ $$$$\:\:\:\:\:\:\:\:\:=+\left(\mathrm{coefficient}\:\mathrm{of}\:{x}\right) \\ $$$$\left({a}−{d}\right)\left({a}\right)+{a}\left({a}+{d}\right)+\left({a}−{d}\right)\left({a}+{d}\right)=\mathrm{11} \\ $$$$\mathrm{2}{a}^{\mathrm{2}} +{a}^{\mathrm{2}} −{d}^{\mathrm{2}} =\mathrm{11} \\ $$$$\mathrm{3}{a}^{\mathrm{2}} −{d}^{\mathrm{2}} =\mathrm{11} \\ $$$${d}^{\mathrm{2}} =\mathrm{3}×\mathrm{4}−\mathrm{11}=\mathrm{1} \\ $$$$\mathrm{product}\:\mathrm{of}\:\mathrm{roots}=−\left(\mathrm{constant}\:\mathrm{term}\right) \\ $$$${m}=−\left({a}−{d}\right)\left({a}\right)\left({a}+{d}\right) \\ $$$$=−\left({a}^{\mathrm{2}} −{d}^{\mathrm{2}} \right){a} \\ $$$$=−\left(\mathrm{4}−\mathrm{1}\right)×\mathrm{6} \\ $$

Commented by Don08q last updated on 24/May/20

Please check again. a will be −2

$$\mathrm{Please}\:\mathrm{check}\:\mathrm{again}.\:{a}\:\mathrm{will}\:\mathrm{be}\:−\mathrm{2} \\ $$

Commented by prakash jain last updated on 24/May/20

Corrected. forgot to divide

$$\mathrm{Corrected}.\:\mathrm{forgot}\:\mathrm{to}\:\mathrm{divide} \\ $$

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