We have learnt in Class-IX, how to factorise quadratic polynomials by splitting their middle
terms. We shall use this knowledge for finding the roots of a quadratic equation. Let us see.
Example-3. Find the roots of the equation 2x
2 – 5x + 3 = 0, by factorisation.
Solution : Let us first split the middle term. Recall that if ax
2 + bx + c is a quadratic
polynomial then to split the middle term we have to find two numbers p and q such that p + q =
b and p × q = a × c. So to split the middle term of 2x
2 – 5x + 3, we have to find two numbers
p and q such that p + q = –5 and p × q = 2 × 3 = 6.
For this we have to list out all possible pairs of factors of 6. They are (1, 6), (–1, –6);
(2, 3); (–2, –3). From the list it is clear that the pair (–2, –3) will satisfy our condition
p + q = –5 and p × q = 6.
The middle term ‘–5x’ can be written as ‘–2x – 3x’.
So, 2x 2 – 5x + 3 = 2x
2 – 2x – 3x + 3 = 2x (x – 1) –3(x – 1) = (2x – 3)(x – 1) Now, 2x
2 – 5x + 3 = 0 can be rewritten as (2x – 3)(x – 1) = 0. So, the values of x for 2x
2 – 5x + 3 = 0 are the same for (2x – 3)(x – 1) = 0, i.e., either 2x – 3 = 0 or x – 1 = 0. Now, 2x – 3 = 0 gives x = 3 2 and x – 1 = 0 gives x = 1.
So, x = 3 2 and x = 1 are the solutions of the equation.
In other words, 1 and 3 2 are the roots of the equation 2x
2 – 5x + 3 = 0.

Note that we have found the roots of 2x 2 – 5x + 3 = 0 by factorising 2x 2 – 5x + 3 into two linear factors and equating each factor to zero.