Question

Solve the following equation by factorization. $6x - \dfrac{2}{x} = 1$

Answer 1

Hint: First, we need to know about the concept of the quadratic equations, which is in the form of at most degree two variables, like the general quadratic equation is $a{x^2} + bx + c = 0$
Since we need to convert the given equation in some form in generalized form then it is easy to solve the given in the factorization method.

Complete step by step answer:
Since from given that $6x - \dfrac{2}{x} = 1$
Now by the multiplication operation, multiply the variable $x$ on the both sides we get $6x - \dfrac{2}{x} = 1 \Rightarrow 6{x^2} - 2 = x$
Now convert all the values in the left side then we get $6{x^2} - x - 2 = 0$ while changing the values on the equals to, the sign of the values or the numbers will change.
Hence the converted equation is in the form of quadratic like $a{x^2} + bx + c = 0$
Now making use of the addition and subtraction we have $ - x = - 4x + 3x$ then we have $6{x^2} - x - 2 = 0 \Rightarrow 6{x^2} - 4x + 3x - 2 = 0$
Now taking the common values from the first two and last two values then we get $2x(3x - 2) + 1(3x - 2) = 0$
Again, taking the common values then we get $(3x - 2)(2x + 1) = 0$ which means either $3x - 2 = 0$ or $2x + 1 = 0$
Hence, we get $x = \dfrac{2}{3}$ or $x = - \dfrac{1}{2}$ which is the required value.

Note:
Also, note that in the quadratic $a{x^2} + bx + c = 0$ the $a$ will never be zero if $a = 0$ then $bx + c = 0$ turns into a linear one-degree equation.
We also make use of the basic operations, The addition is the sum of given two or more than two numbers, or variables and in addition, if we sum the two or more numbers then we obtain a new frame of the number will be found, also in subtraction which is the minus of given two or more than two numbers, but here comes with the condition that in subtraction the greater number sign represented in the number will stay constant example $2 - 3 = - 1$
The other two operations are multiplication and division operations.
Since multiplicand refers to the number multiplied. Also, a multiplier refers to multiplying the first number. Have a look at an example; while multiplying $5 \times 7$the number $5$ is called the multiplicand and the number $7$ is called the multiplier. Like $2 \times 3 = 6$ or which can be also expressed in the form of $2 + 2 + 2(3times)$
The process of the inverse of the multiplication method is called division. Like $x \times y = z$ is multiplication thus the division sees as $x = \dfrac{z}{y}$. Like $3x - 2 = 0 \Rightarrow x = \dfrac{2}{3}$
Hence using simple operations, we solved the given problem.