Question

Solve:
\[{{3}^{2x+4}}+1={{2.3}^{x+2}}\]

Answer 1

Hint: To solve this question we will firstly take \[2\] common from the power of \[3\] which is present in the left hand side. After that we will assume that \[{{3}^{x+2}}\]is equal to some quantity. And then using the basic identity, try to simplify the given equation. By these steps we will get the required answer.

Complete step by step answer:
This is one of the basic questions of mathematics. In the given question ‘.’ is used as the symbol of multiplication. So do not get confused with the multiplication symbol and from the decimal number.
So the given equation is in the form,
\[{{3}^{2x+4}}+1=2\times {{3}^{x+2}}\]
Let us try to simplify the given equation. Firstly take \[2\] common from the power of \[3\] which is present in the left hand side, by doing so we will get
\[\Rightarrow {{3}^{2(x+2)}}+1=2\times {{3}^{x+2}}\]
Assume that \[{{3}^{x+2}}\]is equal to some variable let us say \[A\], so substitute this in the given equation and we will get
\[\Rightarrow {{A}^{2}}+1=2\times A\]
\[\Rightarrow {{A}^{2}}+1-2A=0\]
And we know that, the given equation can be represented as
\[\Rightarrow {{(A-1)}^{2}}=0\]
We can represent this by using the identity, \[{{(a-b)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\].
Now to get the value of \[A\], apply square root on both sides. After applying this we will get
\[\Rightarrow A-1=0\]
Adding \[1\]on both the sides of the equation, we will get
\[\Rightarrow A=1\]
And we previously assumed that \[{{3}^{x+2}}=A\]
We find the value of \[A\] as \[1\], so substituting this value in the above equation we will get
\[\Rightarrow {{3}^{x+2}}=1\]
We know that any base with the power of zero (\[0\]), is always \[1\]. We are only doing this, because it will be easy for us to further solve the equation. So we can represent the above equation as
\[\Rightarrow {{3}^{x+2}}={{3}^{0}}\]
Now the base of the right hand side and left hand side are equal. So we can compare the powers of both sides, we will get
\[\Rightarrow x+2=0\]
If we subtract \[2\]on both the sides, we will get
\[\Rightarrow x=-2\]
So we can conclude that the value for \[x\] is \[-2\]. So \[-2\] is our final answer.

Note:
In these types of questions our basics of mathematics should be strong. And for this equation if we put the value of \[x\] as \[-2\] then the result of the left hand side and right hand side both will be equal. By doing so we can verify our final answer, whether it is correct or not.