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 68234 by Mikael last updated on 07/Sep/19

Commented by kaivan.ahmadi last updated on 07/Sep/19

3^(x+(1/2)) +3^(x−(1/2)) =4^x +2^(2x−1) ⇒ 3^(x−(1/2)) (3+1)=2^(2x−1) (2+1)⇒ (2^(2x−1) /3^(x−(1/2)) )=(4/3)⇒(2^(2x−1) /4)=(3^(x−(1/2)) /3)⇒2^(2x−3) =3^(x+(1/2)) ⇒ { ((2x−3=0⇒x=(3/2))),((x+(1/2)=0⇒x=−(1/2))) :} x=(3/2) is answer

$$\mathrm{3}^{{x}+\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{3}^{{x}−\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{4}^{{x}} +\mathrm{2}^{\mathrm{2}{x}−\mathrm{1}} \Rightarrow \\ $$$$\mathrm{3}^{{x}−\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{3}+\mathrm{1}\right)=\mathrm{2}^{\mathrm{2}{x}−\mathrm{1}} \left(\mathrm{2}+\mathrm{1}\right)\Rightarrow \\ $$$$\frac{\mathrm{2}^{\mathrm{2}{x}−\mathrm{1}} }{\mathrm{3}^{{x}−\frac{\mathrm{1}}{\mathrm{2}}} }=\frac{\mathrm{4}}{\mathrm{3}}\Rightarrow\frac{\mathrm{2}^{\mathrm{2}{x}−\mathrm{1}} }{\mathrm{4}}=\frac{\mathrm{3}^{{x}−\frac{\mathrm{1}}{\mathrm{2}}} }{\mathrm{3}}\Rightarrow\mathrm{2}^{\mathrm{2}{x}−\mathrm{3}} =\mathrm{3}^{{x}+\frac{\mathrm{1}}{\mathrm{2}}} \Rightarrow \\ $$$$\begin{cases}{\mathrm{2}{x}−\mathrm{3}=\mathrm{0}\Rightarrow{x}=\frac{\mathrm{3}}{\mathrm{2}}}\\{{x}+\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{0}\Rightarrow{x}=−\frac{\mathrm{1}}{\mathrm{2}}}\end{cases} \\ $$$${x}=\frac{\mathrm{3}}{\mathrm{2}}\:{is}\:{answer} \\ $$

Commented by Mikael last updated on 07/Sep/19

thanks

$${thanks} \\ $$

Commented by Prithwish sen last updated on 08/Sep/19

2^(2x) =a 3^x =b a−(b/(√3)) = (√3)b−(a/2) ((3a)/2)= ((4b)/(√3)) (a/b) = (2^3 /3^(3/2) ) ∴ 2x=3⇒x=(3/2)

$$\mathrm{2}^{\mathrm{2x}} =\mathrm{a}\:\:\:\mathrm{3}^{\mathrm{x}} =\mathrm{b} \\ $$$$\mathrm{a}−\frac{\mathrm{b}}{\sqrt{\mathrm{3}}}\:=\:\sqrt{\mathrm{3}}\mathrm{b}−\frac{\mathrm{a}}{\mathrm{2}} \\ $$$$\frac{\mathrm{3a}}{\mathrm{2}}=\:\frac{\mathrm{4b}}{\sqrt{\mathrm{3}}} \\ $$$$\frac{\mathrm{a}}{\mathrm{b}}\:=\:\frac{\mathrm{2}^{\mathrm{3}} }{\mathrm{3}^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$$$\therefore\:\mathrm{2x}=\mathrm{3}\Rightarrow\mathrm{x}=\frac{\mathrm{3}}{\mathrm{2}}\: \\ $$

Commented by Rasheed.Sindhi last updated on 09/Sep/19

Sir kaivan ahmadi I don′t understand 2^(2x−3) =3^(x+(1/2)) ⇒^(?) { ((2x−3=0⇒x=(3/2))),((x+(1/2)=0⇒x=−(1/2))) :}

$$\mathrm{Sir}\:\mathrm{kaivan}\:\mathrm{ahmadi}\:\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{understand} \\ $$$$\mathrm{2}^{\mathrm{2}{x}−\mathrm{3}} =\mathrm{3}^{{x}+\frac{\mathrm{1}}{\mathrm{2}}} \overset{?} {\Rightarrow}\begin{cases}{\mathrm{2}{x}−\mathrm{3}=\mathrm{0}\Rightarrow{x}=\frac{\mathrm{3}}{\mathrm{2}}}\\{{x}+\frac{\mathrm{1}}{\mathrm{2}}=\mathrm{0}\Rightarrow{x}=−\frac{\mathrm{1}}{\mathrm{2}}}\end{cases} \\ $$

Commented by kaivan.ahmadi last updated on 13/Sep/19

since 2^0 =3^0 otherwise this equality is not true.

$${since}\:\mathrm{2}^{\mathrm{0}} =\mathrm{3}^{\mathrm{0}} \\ $$$${otherwise}\:{this}\:{equality}\:{is}\:{not}\:{true}. \\ $$

Answered by Rasheed.Sindhi last updated on 09/Sep/19

4^x +4^(x−(1/2)) =3^(x+(1/2)) +3^(x−(1/2)) 4^(x−(1/2)) (4^(1/2) +1)=3^(x−(1/2)) (3+1) 4^(x−(1/2)−1) =3^(x−(1/2)−1) (4^(x−(3/2)) /3^(x−(3/2)) )=1 ((4/3))^(x−(3/2)) =((4/3))^0 x−(3/2)=0 x=(3/2)

$$\mathrm{4}^{\mathrm{x}} +\mathrm{4}^{\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{3}^{\mathrm{x}+\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{3}^{\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$\mathrm{4}^{\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{4}^{\frac{\mathrm{1}}{\mathrm{2}}} +\mathrm{1}\right)=\mathrm{3}^{\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}} \left(\mathrm{3}+\mathrm{1}\right) \\ $$$$\mathrm{4}^{\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{1}} =\mathrm{3}^{\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}−\mathrm{1}} \\ $$$$\frac{\mathrm{4}^{\mathrm{x}−\frac{\mathrm{3}}{\mathrm{2}}} }{\mathrm{3}^{\mathrm{x}−\frac{\mathrm{3}}{\mathrm{2}}} }=\mathrm{1} \\ $$$$\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\mathrm{x}−\frac{\mathrm{3}}{\mathrm{2}}} =\left(\frac{\mathrm{4}}{\mathrm{3}}\right)^{\mathrm{0}} \\ $$$$\mathrm{x}−\frac{\mathrm{3}}{\mathrm{2}}=\mathrm{0} \\ $$$$\mathrm{x}=\frac{\mathrm{3}}{\mathrm{2}} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com