Solve the following equation:
${x^2} - 2x - 8 = 0$

Hint: Factorize the quadratic equation by splitting the middle term into two different terms. Then take out the common factors from the first two terms and last two terms to get the equation in factored form. Finally put each of the factors to zero to determine the roots of the quadratic equation.

Complete step-by-step answer:
According to the question, we have been given a quadratic equation and we have to solve it and find its roots.
The given equation is:
$ \Rightarrow {x^2} - 2x - 8 = 0$
We will use factorization methods to solve this equation. So if we split the middle term into two terms such that the product of their coefficients is equal to the product of first and last coefficients of the quadratic equation, we’ll get:
 $ \Rightarrow {x^2} - 4x + 2x - 8 = 0$
Taking $x$ common from first two terms and 2 common from last two terms, we’ll get:
$ \Rightarrow x\left( {x - 4} \right) + 2\left( {x - 4} \right) = 0$
Simplifying it further, we’ll get:
$ \Rightarrow \left( {x + 2} \right)\left( {x - 4} \right) = 0$
Putting both the factors equal to zero simultaneously, we’ll get:
$
   \Rightarrow \left( {x + 2} \right) = 0{\text{ or }}\left( {x - 4} \right) = 0 \\
   \Rightarrow x = - 2{\text{ or }}x = 4 \\
 $
Thus the two roots of the given quadratic equation are $x = - 2$ and $x = 4$ respectively.
Additional Information:
For a quadratic equation to have real roots, its discriminant must be greater than or equal to zero.
Consider the given quadratic equation:
$ \Rightarrow y = a{x^2} + bx + c$
Discriminant of this quadratic equation is $D = {b^2} - 4ac$
Thus for the quadratic equation to have real roots, following condition must satisfy:
$
   \Rightarrow D \geqslant 0{\text{ or}} \\
   \Rightarrow {b^2} - 4ac \geqslant 0 \\
 $

Note:
If we are facing any difficulty solving a quadratic equation using factorization method, we can also use a direct formula to find its roots. Let the quadratic equation be:
$ \Rightarrow y = a{x^2} + bx + c$
The formula to determine its roots is:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$