Question

Solve the following quadratic equation: ${4^{\left( {x - 1} \right)}} - 3 \cdot {2^{\left( {x - 1} \right)}} + 2 = 0$.

Answer 1

Hint: For solving this type of question, we should always think of making the equation in such a way that it follows some formula or identities. Then we can easily solve it. In this question we will assume ${2^{\left( {x - 1} \right)}} = y$ and substituting the values in the equation we will have the value for the variables.

Complete step-by-step answer:
So we have the equation given ${4^{\left( {x - 1} \right)}} - 3 \cdot {2^{\left( {x - 1} \right)}} + 2 = 0$ .
Let us assume ${2^{\left( {x - 1} \right)}} = y$ , then on substituting into the equation we will get the equation as
$ \Rightarrow {y^2} - 3y + 2 = 0$
For doing the factorization, we will use the method of mid-term splitting, so by splitting the midterm we will get the equation as
$ \Rightarrow {y^2} - 2y - y + 2 = 0$
Now on taking the common, we get
$ \Rightarrow y\left( {y - 2} \right) - 1\left( {y - 2} \right) = 0$
And on solving, it will be equal to
$ \Rightarrow \left( {y - 2} \right)\left( {y - 1} \right) = 0$
And on solving for the values of $y$ , we will get the value as
$ \Rightarrow y = 1,2$
Since we have two values of $y$ , so there will be two cases and we will solve them separately.
If $y = 1$
Then, on substituting the values in the equation ${2^{\left( {x - 1} \right)}} = y$ , we get
$ \Rightarrow {2^{\left( {x - 1} \right)}} = 1$
And the above equation can also be written as,
$ \Rightarrow {2^{\left( {x - 1} \right)}} = {2^0}$
Since the base of the power is the same. Therefore, the power will be equal to each other.
Hence, $x - 1 = 0$
And on solving, we will get
$ \Rightarrow x = 1$
If $y = 2$
Then, on substituting the values in the equation ${2^{\left( {x - 1} \right)}} = y$ , we get
$ \Rightarrow {2^{\left( {x - 1} \right)}} = 2$
And the above equation can also be written as,
$ \Rightarrow {2^{\left( {x - 1} \right)}} = {2^1}$
Since the base of power is the same. Therefore, the power will be equal to each other.
Hence, $x - 1 = 1$
And on solving, we will get
$ \Rightarrow x = 2$
Hence, the value of $x$ will be $\left( {1,2} \right)$ .

Note: Here, in this question, we can see that the equation is easily solved by using the simple mid-term splitting techniques. But sometimes we get the complex equation than for solving those equations we will use the formula $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ and from this, we will get the value for $x$ .