Answer:
The value of x is - 2.
Step-by-step-explanation:
The given equation is
[tex]\displaystyle{\sf\:3^{2x\:+\:4}\:+\:1\:=\:2\:.\:3^{x\:+\:2}}[/tex]
We have to find the value of x.
Now,
[tex]\displaystyle{\implies\sf\:3^{2x\:+\:4}\:+\:1\:=\:2\;\times\:3^{x\:+\:2}}[/tex]
[tex]\displaystyle{\implies\sf\:3^{2\:(x\:+\:2\:)}\:+\:1\:=\:2\;\times\:3^{x\:+\:2}}[/tex]
We know that,
[tex]\displaystyle{\boxed{\green{\sf\:a^{mn}\:=\:(\:a^m\:)^n\:}}}[/tex]
[tex]\displaystyle{\implies\sf\:3^{(\:x\:+\:2\:)^2}\:+\:1\:=\:2\;\times\:3^{x\:+\:2}}[/tex]
[tex]\displaystyle{\boxed{\blue{\sf\:a^{m^n}\:=\:(\:a^m\:)^n\:}}}[/tex]
[tex]\displaystyle{\implies\sf\:(\:3^{x\:+\:2}\:)^2\:+\:1\:=\:2\:\times\:3^{x\:+\:2}}[/tex]
[tex]\displaystyle{\sf\:Let\:3^{x\:+\:2}\:=\:p}[/tex]
[tex]\displaystyle{\implies\sf\:p^2\:+\:1\:=\:2\:\times\:p}[/tex]
[tex]\displaystyle{\implies\sf\:p^2\:-\:2p\:+\:1\:=\:0}[/tex]
[tex]\displaystyle{\implies\sf\:p^2\:-\:2\:\times\:p\:\times\:1\:+\:1^2\:=\:0}[/tex]
a² - 2ab + b² = ( a - b )²
[tex]\displaystyle{\implies\sf\:(\:p\:-\:1\:)^2\:=\:0}[/tex]
[tex]\displaystyle{\implies\sf\:p\:-\:1\:=\:0}[/tex]
[tex]\displaystyle{\implies\sf\:p\:=\:1}[/tex]
[tex]\displaystyle{\sf\:3^{x\:+\:2}\:=\:1}[/tex]
[tex]\displaystyle{\boxed{\pink{\sf\:a^0\:=\:1\:}}}[/tex]
[tex]\displaystyle{\implies\sf\:3^{x\:+\:2}\:=\:3^0}[/tex]
As the bases are equal, powers must be equal.
[tex]\displaystyle{\implies\sf\:x\:+\:2\:=\:0}[/tex]
[tex]\displaystyle{\implies\:\underline{\boxed{\red{\sf\:x\:=\:-\:2\:}}}}[/tex]
∴ The value of x is - 2.
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Verified answer
Answer:
The value of x is - 2.
Step-by-step-explanation:
The given equation is
[tex]\displaystyle{\sf\:3^{2x\:+\:4}\:+\:1\:=\:2\:.\:3^{x\:+\:2}}[/tex]
We have to find the value of x.
Now,
[tex]\displaystyle{\sf\:3^{2x\:+\:4}\:+\:1\:=\:2\:.\:3^{x\:+\:2}}[/tex]
[tex]\displaystyle{\implies\sf\:3^{2x\:+\:4}\:+\:1\:=\:2\;\times\:3^{x\:+\:2}}[/tex]
[tex]\displaystyle{\implies\sf\:3^{2\:(x\:+\:2\:)}\:+\:1\:=\:2\;\times\:3^{x\:+\:2}}[/tex]
We know that,
[tex]\displaystyle{\boxed{\green{\sf\:a^{mn}\:=\:(\:a^m\:)^n\:}}}[/tex]
[tex]\displaystyle{\implies\sf\:3^{(\:x\:+\:2\:)^2}\:+\:1\:=\:2\;\times\:3^{x\:+\:2}}[/tex]
We know that,
[tex]\displaystyle{\boxed{\blue{\sf\:a^{m^n}\:=\:(\:a^m\:)^n\:}}}[/tex]
[tex]\displaystyle{\implies\sf\:(\:3^{x\:+\:2}\:)^2\:+\:1\:=\:2\:\times\:3^{x\:+\:2}}[/tex]
[tex]\displaystyle{\sf\:Let\:3^{x\:+\:2}\:=\:p}[/tex]
[tex]\displaystyle{\implies\sf\:p^2\:+\:1\:=\:2\:\times\:p}[/tex]
[tex]\displaystyle{\implies\sf\:p^2\:-\:2p\:+\:1\:=\:0}[/tex]
[tex]\displaystyle{\implies\sf\:p^2\:-\:2\:\times\:p\:\times\:1\:+\:1^2\:=\:0}[/tex]
We know that,
a² - 2ab + b² = ( a - b )²
[tex]\displaystyle{\implies\sf\:(\:p\:-\:1\:)^2\:=\:0}[/tex]
[tex]\displaystyle{\implies\sf\:p\:-\:1\:=\:0}[/tex]
[tex]\displaystyle{\implies\sf\:p\:=\:1}[/tex]
Now,
[tex]\displaystyle{\sf\:3^{x\:+\:2}\:=\:1}[/tex]
We know that,
[tex]\displaystyle{\boxed{\pink{\sf\:a^0\:=\:1\:}}}[/tex]
[tex]\displaystyle{\implies\sf\:3^{x\:+\:2}\:=\:3^0}[/tex]
As the bases are equal, powers must be equal.
[tex]\displaystyle{\implies\sf\:x\:+\:2\:=\:0}[/tex]
[tex]\displaystyle{\implies\:\underline{\boxed{\red{\sf\:x\:=\:-\:2\:}}}}[/tex]
∴ The value of x is - 2.