The-roots-of-the-equation-2x-2-px-q-0-are-2-and-2-Calculate-the-values-of-p-and-q-

Question Number 158829 by MathsFan last updated on 09/Nov/21

The roots of the equation 2x^2 +px+q=0 are 2α+β and α+2β. Calculate the values of p and q

$$\:{The}\:{roots}\:{of}\:{the}\:{equation} \\ $$$$\:\mathrm{2}{x}^{\mathrm{2}} +{px}+{q}=\mathrm{0}\:{are}\:\mathrm{2}\alpha+\beta\:{and} \\ $$$$\:\alpha+\mathrm{2}\beta.\:{Calculate}\:{the}\:{values}\:{of} \\ $$$$\:{p}\:{and}\:{q} \\ $$

Commented by Rasheed.Sindhi last updated on 09/Nov/21

$${Numerical}\:{answers}\:{of}\:{p}\:\&\:{q}\:{are}\:{not} \\ $$$${possible}\:{because}\:{data}\:{is}\:{insufficient}. \\ $$

Commented by MathsFan last updated on 09/Nov/21

$$\mathrm{yeah},\:\mathrm{sure} \\ $$

Answered by ajfour last updated on 09/Nov/21

(2α+β)+(α+2β)=3(α+β)=−(p/2) (2α+β)(α+2β) =2(α+β)^2 +αβ=(q/2) ⇒ (p^2 /(18))+αβ=(q/2) p=−6(α+β) ; q=(p^2 /9)+2αβ

$$\left(\mathrm{2}\alpha+\beta\right)+\left(\alpha+\mathrm{2}\beta\right)=\mathrm{3}\left(\alpha+\beta\right)=−\frac{{p}}{\mathrm{2}} \\ $$$$\left(\mathrm{2}\alpha+\beta\right)\left(\alpha+\mathrm{2}\beta\right) \\ $$$$\:\:\:\:\:=\mathrm{2}\left(\alpha+\beta\right)^{\mathrm{2}} +\alpha\beta=\frac{{q}}{\mathrm{2}} \\ $$$$\Rightarrow\:\:\:\frac{{p}^{\mathrm{2}} }{\mathrm{18}}+\alpha\beta=\frac{{q}}{\mathrm{2}} \\ $$$${p}=−\mathrm{6}\left(\alpha+\beta\right)\:\:\:;\:{q}=\frac{{p}^{\mathrm{2}} }{\mathrm{9}}+\mathrm{2}\alpha\beta \\ $$

Commented by MathsFan last updated on 09/Nov/21

$$\mathrm{thank}\:\mathrm{you} \\ $$

Facebook Tweet Pin

The-roots-of-the-equation-2x-2-px-q-0-are-2-and-2-Calculate-the-values-of-p-and-q-

Leave a Reply Cancel reply