Menu Close

if-a-b-198-what-is-the-largest-integer-root-which-the-equation-x-2-ax-b-0-may-have-

Tinku Tara 0 Comments
Pin




Question Number 203367 by mr W last updated on 17/Jan/24
if a+b=198, what is the largest integer root which the equation x^2 +ax+b=0 may have?
$${if}\:\boldsymbol{{a}}+\boldsymbol{{b}}=\mathrm{198},\:{what}\:{is}\:{the}\:{largest} \\ $$$$\boldsymbol{{integer}}\:{root}\:{which}\:{the}\:{equation} \\ $$$$\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{ax}}+\boldsymbol{{b}}=\mathrm{0}\:{may}\:{have}? \\ $$
Answered by ajfour last updated on 18/Jan/24
x^2 +ax−a=−(a+b)=−k (x−1)^2 +2(x−1)+a(x−1)=−k−1 (x−1+1+(a/2))^2 =−k−1+(1+(a/2))^2 x=−(a/2)±(√((1+(a/2))^2 −(k+1))) x−1=−(1+(a/2)){1+(√(1−(((k+1))/((1+(a/2))^2 )))) } If we let (1+(a/2))→ −∞ x−1=−2(1+(a/2)) x→∞
$${x}^{\mathrm{2}} +{ax}−{a}=−\left({a}+{b}\right)=−{k} \\ $$$$\left({x}−\mathrm{1}\right)^{\mathrm{2}} +\mathrm{2}\left({x}−\mathrm{1}\right)+{a}\left({x}−\mathrm{1}\right)=−{k}−\mathrm{1} \\ $$$$\left({x}−\mathrm{1}+\mathrm{1}+\frac{{a}}{\mathrm{2}}\right)^{\mathrm{2}} =−{k}−\mathrm{1}+\left(\mathrm{1}+\frac{{a}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$${x}=−\frac{{a}}{\mathrm{2}}\pm\sqrt{\left(\mathrm{1}+\frac{{a}}{\mathrm{2}}\right)^{\mathrm{2}} −\left({k}+\mathrm{1}\right)} \\ $$$${x}−\mathrm{1}=−\left(\mathrm{1}+\frac{{a}}{\mathrm{2}}\right)\left\{\mathrm{1}+\sqrt{\mathrm{1}−\frac{\left({k}+\mathrm{1}\right)}{\left(\mathrm{1}+\frac{{a}}{\mathrm{2}}\right)^{\mathrm{2}} }}\:\right\} \\ $$$${If}\:\:{we}\:\:{let}\:\:\left(\mathrm{1}+\frac{{a}}{\mathrm{2}}\right)\rightarrow\:−\infty \\ $$$${x}−\mathrm{1}=−\mathrm{2}\left(\mathrm{1}+\frac{{a}}{\mathrm{2}}\right) \\ $$$${x}\rightarrow\infty \\ $$
Answered by AST last updated on 26/Jan/24
−a=x_1 +x_2 ;b=x_1 x_2 198=a+b=x_1 (x_2 −1)−x_2 ⇒x_1 =((198+x_2 )/(x_2 −1))=1+((199)/(x_2 −1)) As x_2 →1^+ ;x_1 →∞⇒no largest integer root
$$−{a}={x}_{\mathrm{1}} +{x}_{\mathrm{2}} ;{b}={x}_{\mathrm{1}} {x}_{\mathrm{2}} \\ $$$$\mathrm{198}={a}+{b}={x}_{\mathrm{1}} \left({x}_{\mathrm{2}} −\mathrm{1}\right)−{x}_{\mathrm{2}} \Rightarrow{x}_{\mathrm{1}} =\frac{\mathrm{198}+{x}_{\mathrm{2}} }{{x}_{\mathrm{2}} −\mathrm{1}}=\mathrm{1}+\frac{\mathrm{199}}{{x}_{\mathrm{2}} −\mathrm{1}} \\ $$$${As}\:{x}_{\mathrm{2}} \rightarrow\mathrm{1}^{+} ;{x}_{\mathrm{1}} \rightarrow\infty\Rightarrow{no}\:{largest}\:{integer}\:{root} \\ $$

Pin

Leave a Reply

Your email address will not be published. Required fields are marked *