Question

Solve: ${{4}^{x}}-{{5.2}^{x}}+4=0$ to find the values of x.

Answer 1

Hint: Take ${{2}^{x}}=t$ and transform the given equation as, ${{t}^{2}}-5t+4=0$ . Then factorize it to find values of t and hence, find values of x too.

Complete step-by-step answer:

In the question we have been given an equation ${{4}^{x}}-{{5.2}^{x}}+4=0$ such that we have to solve or find values of x.

Now, let’s write the equation

${{4}^{x}}-{{5.2}^{x}}+4=0$

We can further write it as,

${{\left( {{2}^{x}} \right)}^{2}}-{{5.2}^{x}}+4=0$

Now, in the equation given we will take the value of ${{2}^{x}}$ and take it as variable t.

So, the equation which is given will be written as,

${{t}^{2}}-5t+4=0$

Now we will factorize it by using the middle term factor. So, we can write equation as,

${{t}^{2}}-t-4t+4=0$

So,

$t\left( t-1 \right)-4\left( t-1 \right)=0$

Now, we can write it as

$\left( t-4 \right)\left( t-1 \right)=0$

So, for two values of t equation satisfies which is $t=1$ and $t=4$

We know that, ${{2}^{x}}$ was taken and substituted as t, so

${{2}^{x}}=t$

Here, $t=1,4$

So, for some value of x, ${{2}^{x}}$ will be equal to 4 which is achievable when $x=2$.

So, for values which x satisfies is 0, 2.

The value of x is 0, 2.

Note: While solving quadratic equations one can also use formulas like completing the square or using Sridhar Acharya’s formula.