# Factorisation of a Quadratic Trinomial by Splitting the Middle Term

#### definition

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

#### formula

• (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)

#### Example

Factorise: 2x2 + 5x - 18

2x2 + 5x - 18

= 2x2 + 9x - 4x - 18

= x(2x + 9) - 2(2x + 9)

= (2x + 9)(x - 2)

#### Example

Factorise: x2 - 10x + 21.

x2 - 10x + 21

= x2 - 10x + 21

= x2 - 7x - 3x + 21

= x(x - 7) - 3(x - 7)

= (x - 7)(x - 3)

#### Example

Find the factors of 2y2 - 4y - 30.

2y2 - 4y - 30

= 2(y2 - 2y - 15)      .....(taking out the common factor 2)

= 2(y2 - 5y + 3y - 15)

= 2[y(y - 5) + 3(y - 5)]

= 2(y - 5)(y + 3).

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