Question

Solve the equation ${5^{2x}} - {24.5^x} - 25 = 0$ find the number of roots.

Answer 1

Hint: Here we are asked to solve the given equation and find the number of roots. For that, we will first try to reduce the given equation by the substitution method. Then we will find the roots of that reduced equation from that we can find the number roots the given equation has.

Complete step-by-step answer:
It is given that the equation is ${5^{2x}} - {24.5^x} - 25 = 0$ . We aim to find the number of roots of this equation.
To find the number of roots of the given equation we will try to reduce the given equation since it has unknown variables in its powers.
Consider the given equation ${5^{2x}} - {24.5^x} - 25 = 0$ let us rewrite the given equation for our convenience.
$ \Rightarrow {\left( {{5^x}} \right)^2} - 24\left( {{5^x}} \right) - 25 = 0$
Now we can see that the term ${5^x}$ is common in the two terms. Now let us substitute ${5^x} = t$ in the above equation.
$ \Rightarrow {t^2} - 24t - 25 = 0$
We can see that the given equation is reduced to a quadratic equation. We know that the quadratic equation will have two roots but let us check whether it has two roots or not.
Let us solve the above quadratic equation.
$ \Rightarrow {t^2} - 25t + t - 25 = 0$
$ \Rightarrow t\left( {t + 1} \right) - 25\left( {t + 1} \right) = 0$
$ \Rightarrow \left( {t + 1} \right)\left( {t - 25} \right) = 0$
$ \Rightarrow t = - 1,t = 25$
Now we got the values of $t$ . Now let us re-substitute the values of $t$ in ${5^x}$ .
For $t = 25$ , ${5^x} = t = 25$ on solving this we get $x = 2$ .
For $t = - 1$ , ${5^x} = t = - 1$ on solving this we get an imaginary root so we will neglect this value of $t$ ,
Thus, the given equation has only one real root.

Note: In the above problem, we are given an equation that has an unknown variable in its power. It is difficult to find the roots directly solving those equations. So, we have reduced it by using the substitution method which made the given equation simpler to solve. After solving the equation for the substitute, we have to find the original roots by substituting them again in the original term to find their original roots.