Solve for x, $2{x^2} - x = 0$

Hint:Check for the highest power of the variable. The highest power determines the number of solutions for the equation. In this case, the highest power is 2 and hence you’ll get 2 solutions for the quadratic equation.

Complete step by step solution:
The given equation has a highest power of 2 on variable “x” and thus revealing that the equation will give us 2 solutions.
The standard quadratic equation looks like,
$a{x^2} + bx + c = 0$
Comparing the standard equation to that of the question, we see that
a = 2, b = -1 and c = 0.
Thus, we can rewrite the equation in the question as,
$2{x^2} - x + 0 = 0$
If we check the question for “Mid-term factorization”, we see that it is not possible.
We take an easier approach.
$2{x^2} - x = 0$
$ \Rightarrow 2{x^2} = x$…… (Taking x from LHS to RHS)
\[ \Rightarrow \dfrac{{2{x^2}}}{x} = \dfrac{x}{x}\]…… (Dividing throughout by x)
$ \Rightarrow 2x = 1$
$ \Rightarrow \dfrac{{2x}}{2} = \dfrac{1}{2}$……. (Dividing throughout by 2)
$ \Rightarrow x = \dfrac{1}{2}$
This above method gives us only one value of x. But we know that there are2 values that are to be found.
If we take the following approach, 2 values are found.
$2{x^2} - x = 0$
$ \Rightarrow (2x - 1)x = 0$
This means that x=0 or $(2x - 1) = 0$
$ \Rightarrow x = 0\;or\;x = \dfrac{1}{2}$

Note: Incase of imaginary values, this method is very cumbersome and hence should not be used. In such a case, you can use the formula, $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$to find the factors. This formula can be used for any kind of quadratic equation arranged as per the standard quadratic equation. It reduces the time required to solve more complex equations.