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](https://www.tinkutara.com/question/Q6477.png)
$${Given}\:{equation}:\:{x}^{\mathrm{2}} −{px}+\mathrm{8}=\mathrm{0} \\ $$$$\:\:\:\:\:{Sum}\:{of}\:{the}\:{roots}\:=−\frac{{coefficient}\:{of}\:{x}}{{coefficient}\:{of}\:{x}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=−\frac{−{p}}{\mathrm{1}}={p} \\ $$$$\:\:\:\:\:\:{And}\:{this}\:{is}\:{equal}\:{to}\:{a}+\left({a}+\mathrm{2}\right)=\mathrm{2}{a}+\mathrm{2} \\ $$$$\:\:\:\:\:\:{Hence}\:\:\:\:\:{p}=\mathrm{2}{a}+\mathrm{2} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:{Product}\:{of}\:{the}\:{roots}=\frac{{constant}}{{coefficient}\:{of}\:{x}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{8}}{\mathrm{1}}=\mathrm{8} \\ $$$$\:\:\:\:\:\:\:\:\:{And}\:{this}\:{is}\:{equal}\:{to}\:{a}\left({a}+\mathrm{2}\right) \\ $$$$\:\:\:\:\:\:\:\:\:{Hence}\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}\left({a}+\mathrm{2}\right)=\mathrm{8} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}^{\mathrm{2}} +\mathrm{2}{a}−\mathrm{8}=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({a}+\mathrm{4}\right)\left({a}−\mathrm{2}\right)=\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}=−\mathrm{4}\:\:\mid\:\:\:{a}=\mathrm{2} \\ $$$${If}\:\:\:{a}=−\mathrm{4}\Rightarrow{p}=\mathrm{2}\left(−\mathrm{4}\right)+\mathrm{2}=−\mathrm{6}\:\:\:\:\:\:\left[\because\:{p}=\mathrm{2}{a}+\mathrm{2}\right] \\ $$$${If}\:\:{a}=\mathrm{2}\Rightarrow{p}=\mathrm{2}\left(\mathrm{2}\right)+\mathrm{2}=\mathrm{6} \\ $$$${Hence}\:{p}=\pm\mathrm{6} \\ $$
Commented by sanusihammed last updated on 28/Jun/16
$${Thanks}\:{so}\:{much} \\ $$