Question

Find the roots of the following equation using factorisation method.
                           ${x^2} - 5x + 6 = 0$

Answer 1

Hint: Here we will proceed by splitting the middle term. Then we will take the common terms out and hence we will get the required roots of the given quadratic equation i.e. ${x^2} - 5x + 6 = 0$.

Complete step-by-step answer:
Here we are given with the quadratic equation i.e. ${x^2} - 5x + 6 = 0$
So as we know that, it is a quadratic equation and it will have two roots.
Hence we need to find numbers in this method who have a sum of 5 and a product of 6.
Also looking to the requirement, we can have the following number of possibilities for two numbers to have a sum of 5 and a product 6. Which could be-
0 and 5, here the sum is 5 but the product is 0, hence this is not the one we are looking for.
1 and 4, here the sum is 5 but the product is 4, hence it is also not the one we are looking for.
2 and 3, here the sum is 5 and the product is 6, so these are the two numbers which we were looking for.
Hence now rewrite the above equation in the terms of the two numbers 2 and 3-
$ \Rightarrow {x^2} - 2x - 3x + 6 = 0$
Now here we can see that the first two terms have x common in them and also in other two terms 3 is common so we will take both the common terms out.
$ \Rightarrow x\left( {x - 2} \right) - 3\left( {x - 2} \right)$
Here we come with two possibilities as –
$\left( {x - 2 = 0} \right){\text{ or }}\left( {x - 3 = 0} \right)$
From here we have the two roots of the above equation i.e. ${x^2} - 5x + 6 = 0$ which are x=2 and x=3.
So, the answer is x = 2 and x = 3.

Note: In order to solve this type of question, we must know the method of factorization. In this method we will find the factors so that the middle part splits into two parts i.e. splitting the middle terms into the quadratic equation i.e. ${x^2} - 5x + 6 = 0$. Thus, we get our desired answer.