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5^(x+1)+5^(2-x)=5^3+1

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5^(x+1)+5^(2-x)=5^3+1

Posted by Somya Khandelwal 6 years, 6 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 6 months ago

Given,
5^(x+1) + 5^{(2 - x)} = 5³ +1
{tex}\Rightarrow{/tex} 5^x.5 + 5².5^-x = 125 + 1

{tex}\Rightarrow{/tex} 5^x.5 + {tex}\frac { 25 } { 5 ^ { x } }{/tex} = 126

Let {tex}y =5^x{/tex}
{tex}\Rightarrow{/tex} 5y + {tex}\frac { 25 } { y }{/tex} = 126

{tex}\Rightarrow5y^2+25=126y{/tex}
{tex}\Rightarrow{/tex} 5y²- 126y + 25 = 0

Factorise the equation,

{tex}\Rightarrow{/tex} 5y²- 125y - y + 25 = 0
{tex}\Rightarrow{/tex} 5y(y - 25) -1 (y - 25) = 0

{tex}\Rightarrow{/tex}(y - 25) (5y - 1) = 0
{tex}\Rightarrow{/tex} y - 25 = 0 or 5y - 1 = 0
{tex}\Rightarrow{/tex} y = 25 or y = 1/5

Now,
{tex}\Rightarrow{/tex} 5^x = 25 or 5^x = 1/5

{tex}\Rightarrow{/tex} 5^x = 5² or 5^x = 5^-1
{tex}\Rightarrow{/tex} x = 2 or x = -1.

0Thank You

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CBSE > Class 10 > Mathematics