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Question Number 6476 by sanusihammed last updated on 28/Jun/16

Answered by Rasheed Soomro last updated on 28/Jun/16

Given equation: x^2 −px+8=0       Sum of the roots =−((coefficient of x)/(coefficient of x^2 ))                                            =−((−p)/1)=p        And this is equal to a+(a+2)=2a+2        Hence     p=2a+2            Product of the roots=((constant)/(coefficient of x^2 ))                                                      =(8/1)=8           And this is equal to a(a+2)           Hence              a(a+2)=8                                        a^2 +2a−8=0                                         (a+4)(a−2)=0                                           a=−4  ∣   a=2  If   a=−4⇒p=2(−4)+2=−6      [∵ p=2a+2]  If  a=2⇒p=2(2)+2=6  Hence p=±6

$${Given}\:{equation}:\:{x}^{\mathrm{2}} −{px}+\mathrm{8}=\mathrm{0} \\ $$$$\:\:\:\:\:{Sum}\:{of}\:{the}\:{roots}\:=−\frac{{coefficient}\:{of}\:{x}}{{coefficient}\:{of}\:{x}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=−\frac{−{p}}{\mathrm{1}}={p} \\ $$$$\:\:\:\:\:\:{And}\:{this}\:{is}\:{equal}\:{to}\:{a}+\left({a}+\mathrm{2}\right)=\mathrm{2}{a}+\mathrm{2} \\ $$$$\:\:\:\:\:\:{Hence}\:\:\:\:\:{p}=\mathrm{2}{a}+\mathrm{2} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{Product}\:{of}\:{the}\:{roots}=\frac{{constant}}{{coefficient}\:{of}\:{x}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{8}}{\mathrm{1}}=\mathrm{8} \\ $$$$\:\:\:\:\:\:\:\:\:{And}\:{this}\:{is}\:{equal}\:{to}\:{a}\left({a}+\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:{Hence}\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}\left({a}+\mathrm{2}\right)=\mathrm{8} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} +\mathrm{2}{a}−\mathrm{8}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({a}+\mathrm{4}\right)\left({a}−\mathrm{2}\right)=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}=−\mathrm{4}\:\:\mid\:\:\:{a}=\mathrm{2} \\ $$$${If}\:\:\:{a}=−\mathrm{4}\Rightarrow{p}=\mathrm{2}\left(−\mathrm{4}\right)+\mathrm{2}=−\mathrm{6}\:\:\:\:\:\:\left[\because\:{p}=\mathrm{2}{a}+\mathrm{2}\right] \\ $$$${If}\:\:{a}=\mathrm{2}\Rightarrow{p}=\mathrm{2}\left(\mathrm{2}\right)+\mathrm{2}=\mathrm{6} \\ $$$${Hence}\:{p}=\pm\mathrm{6} \\ $$

Commented by sanusihammed last updated on 28/Jun/16

Thanks so much

$${Thanks}\:{so}\:{much} \\ $$

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