Question

How do you solve the given equation: $\dfrac{1}{x}-\dfrac{1}{2x}=2x$?

Answer 1

Hint: We start solving the problem by simplifying the L.H.S (Left Hand Side) of the given equation by performing different mathematical operations. We then cross multiply the numerator and denominator on both sides to get the quadratic equation in x. We then make use of the quadratic formula that the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ are defined as $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ to proceed through the problem. We then make the necessary calculations to find the roots of the obtained quadratic equation and then neglect the values at which the given function is not defined to get the required answer.

Complete step-by-step answer:
According to the problem, we are asked to solve the given equation: $\dfrac{1}{x}-\dfrac{1}{2x}=2x$.
We have given the equation: $\dfrac{1}{x}-\dfrac{1}{2x}=2x$ ---(1).
Now, let us simplify the terms present in L.H.S (Left Hand Side) of the equation (1).
$\Rightarrow \dfrac{2-1}{2x}=2x$.
$\Rightarrow \dfrac{1}{2x}=2x$ ---(2).
Let us cross multiply the numerator and denominator of both sides in equation (2).
$\Rightarrow 1=2x\times 2x$.
$\Rightarrow 1=4{{x}^{2}}$.
$\Rightarrow 4{{x}^{2}}-1=0$ ---(3).
We know that the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ are defined as $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$, which is also known as quadratic formula.
Now, let us compare the given quadratic equation $4{{x}^{2}}-1=0$ with $a{{x}^{2}}+bx+c=0$. So, we get $a=4$, $b=0$, $c=-1$. Now, let us substitute these values in quadratic formula to get the roots of the given quadratic equation $4{{x}^{2}}-1=0$.
So, the roots are $x=\dfrac{-\left( 0 \right)\pm \sqrt{{{\left( 0 \right)}^{2}}-4\left( 4 \right)\left( -1 \right)}}{2\left( 4 \right)}$.
$\Rightarrow x=\dfrac{0\pm \sqrt{0-\left( -16 \right)}}{8}$.
$\Rightarrow x=\dfrac{0\pm \sqrt{16}}{8}$.
$\Rightarrow x=\dfrac{\pm 4}{8}$.
$\Rightarrow x=\dfrac{\pm 1}{2}$.
So, we have found the solution of the given equation $\dfrac{1}{x}-\dfrac{1}{2x}=2x$ as $x=\dfrac{\pm 1}{2}$.
$\therefore $ The solution of the given equation$\dfrac{1}{x}-\dfrac{1}{2x}=2x$ is $x=\dfrac{\pm 1}{2}$.

Note: We should perform each step carefully in order to avoid confusion and calculation mistakes while solving this problem. We can also verify the obtained answer by substituting it in the given equation and checking the values in both L.H.S (Left Hand Side) and R.H.S (Right Hand Side). Similarly, we can expect the problems to find the solution of the given equation: $\dfrac{3x-2}{x-4}=\dfrac{5}{x-4}+\dfrac{1}{x-3}$..