Question

Solve the following equation:
$6\sqrt {\text{x}} = {\text{5}}{{\text{x}}^{ - \dfrac{1}{2}}} - 13$

Answer 1

Hint: To solve the following equation, as there is a square root term in the equation and it is difficult to solve, we take it as a variable and solve the equation. After finding the solution we put the square root term back in the equation to determine its value.
We know a number of the form $\sqrt {\text{a}} $ can be expressed as ${{\text{a}}^{\dfrac{1}{2}}}$.

Complete step by step solution:
Given Data,
$6\sqrt {\text{x}} = {\text{5}}{{\text{x}}^{ - \dfrac{1}{2}}} - 13$

$
   \Rightarrow 6\sqrt {\text{x}} = {\text{5}}{{\text{x}}^{ - \dfrac{1}{2}}} - 13 \\
   \Rightarrow 6\sqrt {\text{x}} = \dfrac{5}{{\sqrt {\text{x}} }} - 13 \\
  {\text{Now we put }}\sqrt {\text{x}} = {\text{t}} \\
   \Rightarrow 6{\text{t}} = \dfrac{5}{{\text{t}}} - 13 \\
   \Rightarrow {\text{6}}{{\text{t}}^2} + 13{\text{t - 5 = 0}} \\
   \Rightarrow {\text{6}}{{\text{t}}^2} - 2{\text{t + }}15{\text{t - 5 = 0}} \\
   \Rightarrow {\text{2t}}\left( {{\text{3t - 1}}} \right) + 5\left( {{\text{3t - 1}}} \right) = 0 \\
   \Rightarrow \left( {{\text{2t + 5}}} \right)\left( {3{\text{t - 1}}} \right) = 0 \\
   \Rightarrow {\text{t = - }}\dfrac{5}{2},\dfrac{1}{3} \\
  {\text{As }}\sqrt {\text{x}} = {\text{t}} \\
   \Rightarrow {\text{x = }}{{\text{t}}^2} \\
   \Rightarrow {\text{x = }}{\left( { - \dfrac{5}{2}} \right)^2},{\left( {\dfrac{1}{3}} \right)^2} \\
   \Rightarrow {\text{x = }}\dfrac{{25}}{4},\dfrac{1}{9} \\
$
Therefore the solutions of the given equation are ${\text{x = }}\dfrac{{25}}{4},\dfrac{1}{9}$.

Note: In order to solve this type of questions the key is to know how to go about in solving this type of questions. We can only achieve this by solving more questions of this type. We generally express the square root or root of any order term as a variable and solve the equation. After we solve it we put the root term back and compute it. This is the traditional way of solving this type of equation.
Any term having a degree off root on it can be expressed as the number to the power of a fraction of the degree of the root, which is as shown in the above. $\sqrt {\text{a}} = {{\text{a}}^{\dfrac{1}{2}}}$.