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

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

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

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

Commented by Don08q last updated on 24/May/20

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

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

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

Commented by Don08q last updated on 24/May/20

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

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

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

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