Factorisation of a Quadratic Trinomial by Splitting the Middle Term

• Quadratic Trinomial: An expression of the form `ax^2 + bx + c` is called a quadratic trinomial.

• (x + a)(x + b) = x² + (a + b)x + ab

Factorisation of a Quadratic Trinomial:

An expression of the form `ax^2 + bx + c` is called a quadratic trinomial.

We know that `(x + a)(x + b) = x^2 + (a + b)x + ab.`

∴ the factors of `x^2 + (a + b)x + ab  "are"  (x + a) and (x + b).`

To find the factors of `x^2 + 5x + 6, "by comparing it with"  x^2 + (a + b)x + ab.`

we get, a + b = 5 and ab = 6. So, let us find the factors of 6 whose sum is 5.

Then writing the trinomial in the form `x^2 + (a + b)x + ab`, find its factors.

`x^2 + 5x + 6 = x^2 + (3 + 2)x + 3 xx 2      ......[x^2 + (a + b)x + ab]`

`= x^2 + 3x + 2x + 2 xx 3`                            .......[Multiply (3 + 2) by x, make two groups of the four terms obtained.]

`= x(x + 3) + 2(x + 3)`

`= (x + 3)(x + 2)`

Factorise: 2x² - 9x + 9.

First, we find the product of the coefficient of the square term and the constant term. Here the product is 2 × 9 = 18.

Now, find factors of 18 whose sum is -9, which is equal to the coefficient of the middle term.

2x² - 9x + 9

= 2x² - 6x - 3x + 9

= 2x(x - 3) - 3(x - 3)

= (x - 3)(2x - 3)

∴ 2x² - 9x + 9 = (x - 3)(2x - 3)