Factorize the Trinomial ax Square Plus bx Plus c

Factorize the trinomial ax square plus bx plus c means ax2 + bx + c.

In order to factorize the expression ax2 + bx + c, we have to find two numbers m and n, such that m + n = b and m × n = ac.

That is we split b into two parts m and n whereas sum m and n = b and product m and n = ac.

Solved examples to factorize the trinomial ax square plus bx plus c (ax^2 + bx + c):

1. Resolve into factors:

(i) 2x2 + 9x + 10

Solution:

The given expression is 2x2 + 9x + 10.

Find two numbers whose sum = 9 and product = (2 × 10) = 20.

Clearly, such numbers are 5 and 4.

Therefore, 2x2 + 9x + 10 = 2x2 + 5x + 4x + 10

= x(2x + 5) + 2(2x + 5)

= (2x + 5)(x + 2).

(ii) 6x2 + 7x - 3

Solution:

The given expression is 6x2 + 7x - 3.

Find two numbers whose sum = 7 and product = 6 × (-3) = -18.

Clearly, such numbers are 9 and -2.

Therefore, 6x2 + 7x - 3 = 6x2 + 9x - 2x - 3

= 3x (2x + 3) -1 (2x + 3)

= (2x + 3)(3x - 1).

2. Factorize the trinomial:

(i) 2m2 +7m + 3

Solution:

The given expression is 2m2 +7m + 3.

Here, the two numbers a and b are such that their sum x + y =7 and their product x × y = 3 × 2 i.e., x × y = 6

Such numbers are 1 to 6

Now, splitting the middle term 7m of the given expression 2m2 + 7m + 3 we get,

= 2m2 + 1m + 6m + 3

= m(2m + 1) + 3(2m + 1)

= (2m +1)(m + 3)

(ii) 3x2 - 4x - 4

Solution:

The given expression is 3x2 - 4x - 4.

Find two numbers whose sum = -4 and product = 3 × (-4) = -12.

Clearly, such numbers are -6 and 2.

Therefore, 3x2 - 4x - 4 = 3x2 - 6x + 2x - 4

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

= (x - 2)(3x + 2).

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