Question

How do you solve $\dfrac{{2x}}{1} = \dfrac{1}{{x - 1}}$?

Answer 1

Hint: Here we will find the value of “x” by using the first step as the cross-multiplication for the above expression. In cross multiplication, the numerator of one side is multiplied with the denominator of the opposite side.

Complete step-by-step answer:
Take the given expression:
$\dfrac{{2x}}{1} = \dfrac{1}{{x - 1}}$
Do cross multiplication, where the numerator of one side is multiplied with the denominator of the opposite side.
$ \Rightarrow 2x(x - 1) = 1(1)$
Multiply the term outside the bracket inside the bracket. Remember when you multiply two positive terms the resultant value is positive and when you multiply one negative and one positive term the resultant value is negative term.
$ \Rightarrow 2{x^2} - 2x = 1$
Move the term from right to the left of the equation. When you move any term from one side to another, the sign also changes. Positive terms become negative.
$ \Rightarrow 2{x^2} - 2x - 1 = 0$
Now, compare the above equation with the standard quadratic equation.
$a{x^2} + bx + c = 0$
Compare the standard equation with the above equation $2{x^2} - 2x - 1 = 0$
$
   \Rightarrow a = 2 \\
   \Rightarrow b = - 2 \\
   \Rightarrow c = - 1 \;
 $
Now, $\Delta = {b^2} - 4ac$
Place the values in the above equation –
$\Delta = {( - 2)^2} - 4(2)( - 1)$
Simplify the above equation-
$
  \Delta = 4 + 8 \\
  \Delta = 12 \;
 $
Take the square root on both sides of the equation.
$ \Rightarrow \sqrt \Delta = \sqrt {12} $
Simplify the above equation-
$ \Rightarrow \sqrt \Delta = 2\sqrt 3 $
Now, the roots can be expressed as
 \[x = \dfrac{{ - b \pm \sqrt \Delta }}{{2a}}\]
Therefore, \[x = \dfrac{{ - b + \sqrt \Delta }}{{2a}}\] or \[x = \dfrac{{ - b - \sqrt \Delta }}{{2a}}\]
Place the known values in the above equation –
 \[x = \dfrac{{ - ( - 2) + 2\sqrt 3 }}{{2(2)}}\] or \[x = \dfrac{{ - ( - 2) - 2\sqrt 3 }}{{2(2)}}\]
Simplify the above equation
 \[x = \dfrac{{2 + 2\sqrt 3 }}{{2(2)}}\] or \[x = \dfrac{{2 - 2\sqrt 3 }}{{2(2)}}\]
Take out common factors from the numerator and the denominator. Common factors from the numerator and the denominator cancel each other.
 \[x = \dfrac{{1 + \sqrt 3 }}{2}\] or \[x = \dfrac{{1 - \sqrt 3 }}{2}\]
This is the required solution.
So, the correct answer is “ \[x = \dfrac{{1 + \sqrt 3 }}{2}\] or \[x = \dfrac{{1 - \sqrt 3 }}{2}\]”.

Note: Always remember that product of two negative terms gives us the positive term. Also remember that the square of negative number or the positive number always gives the positive number. Square is the number multiplied with itself twice.