Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 175715 by zaheen last updated on 05/Sep/22

how is the solution of this qution    (√((x)(x+1)(x+2)(x+3)+1))  when      determinant (((x=50))) determinant ((),())

$${how}\:{is}\:{the}\:{solution}\:{of}\:{this}\:{qution} \\ $$$$ \\ $$$$\sqrt{\left({x}\right)\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\left({x}+\mathrm{3}\right)+\mathrm{1}} \\ $$$${when}\:\:\:\:\:\begin{array}{|c|}{{x}=\mathrm{50}}\\\hline\end{array}\begin{array}{|c|c|}\\\\\hline\end{array} \\ $$

Commented by Frix last updated on 05/Sep/22

(x+n)(x+n+1)(x+n+2)(x+n+3)+1=  =(x^2 +(2n+3)x+n^2 +3n+1)^2

$$\left({x}+{n}\right)\left({x}+{n}+\mathrm{1}\right)\left({x}+{n}+\mathrm{2}\right)\left({x}+{n}+\mathrm{3}\right)+\mathrm{1}= \\ $$$$=\left({x}^{\mathrm{2}} +\left(\mathrm{2}{n}+\mathrm{3}\right){x}+{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{1}\right)^{\mathrm{2}} \\ $$

Answered by ajfour last updated on 05/Sep/22

let  x=t−(3/2)  ⇒  x+3=t+(3/2)  f(t)=(√((t^2 −(9/4))(t^2 −(1/4))+1))  let    t^2 −(5/4)=s  f(s)=(√(s^2 −1+1))=s           =t^2 −(5/4)=(x+(3/2))^2 −(5/4)  f(50)=((103^2 −5)/4)=2651

$${let}\:\:{x}={t}−\frac{\mathrm{3}}{\mathrm{2}}\:\:\Rightarrow\:\:{x}+\mathrm{3}={t}+\frac{\mathrm{3}}{\mathrm{2}} \\ $$$${f}\left({t}\right)=\sqrt{\left({t}^{\mathrm{2}} −\frac{\mathrm{9}}{\mathrm{4}}\right)\left({t}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{4}}\right)+\mathrm{1}} \\ $$$${let}\:\:\:\:{t}^{\mathrm{2}} −\frac{\mathrm{5}}{\mathrm{4}}={s} \\ $$$${f}\left({s}\right)=\sqrt{{s}^{\mathrm{2}} −\mathrm{1}+\mathrm{1}}={s} \\ $$$$\:\:\:\:\:\:\:\:\:={t}^{\mathrm{2}} −\frac{\mathrm{5}}{\mathrm{4}}=\left({x}+\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} −\frac{\mathrm{5}}{\mathrm{4}} \\ $$$${f}\left(\mathrm{50}\right)=\frac{\mathrm{103}^{\mathrm{2}} −\mathrm{5}}{\mathrm{4}}=\mathrm{2651} \\ $$

Commented by Tawa11 last updated on 05/Sep/22

Great sir

$$\mathrm{Great}\:\mathrm{sir} \\ $$

Answered by behi834171 last updated on 05/Sep/22

x(x+3)=x^2 +3x  (x+1)(x+2)=x^2 +3x+2  x^2 +3x=t  ⇒f(t)=(√((t)(t+2)+1))=(√(t^2 +2t+1))=t+1  ⇒f(x)=x^2 +3x+1  ⇒f(x)=50^2 +3×50+1=2651 . ■

$${x}\left({x}+\mathrm{3}\right)={x}^{\mathrm{2}} +\mathrm{3}{x} \\ $$$$\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)={x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{2} \\ $$$${x}^{\mathrm{2}} +\mathrm{3}{x}={t} \\ $$$$\Rightarrow{f}\left({t}\right)=\sqrt{\left({t}\right)\left({t}+\mathrm{2}\right)+\mathrm{1}}=\sqrt{{t}^{\mathrm{2}} +\mathrm{2}{t}+\mathrm{1}}=\boldsymbol{{t}}+\mathrm{1} \\ $$$$\Rightarrow\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{3}\boldsymbol{{x}}+\mathrm{1} \\ $$$$\Rightarrow\boldsymbol{{f}}\left(\boldsymbol{{x}}\right)=\mathrm{50}^{\mathrm{2}} +\mathrm{3}×\mathrm{50}+\mathrm{1}=\mathrm{2651}\:.\:\blacksquare \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com