#### Topic Content:

- Factorization of Quadratic Expressions II
- Quadratic Expression of the form ax
^{2}+ bx + c

- Quadratic Expression of the form ax

### Quadratic Expression of the form ax^{2} + bx + c

#### Factorization of Quadratic Equation by Splitting the Middle Term:

In order to factorise a quadratic expression of the form ax^{2} + bx + c, where \( \scriptsize a \neq 1 \)

Consider the quadratic equation ax^{2} + bx + c = 0

**Step 1:** Multiply the coefficient of **x ^{2}** and the constant term

**c**i.e.

**a**×

**c**=

**ac**

**Step 2:** Now, find two numbers such that their product is equal to ac and sum equals to b.

**Product: **(number 1)(number 2) = ac**Sum:** (number 1) + (number 2) = b

**Step 3 (Fcatorise):** Now, split the middle term using these two numbers,

x^{2} + (number 1)x + (number 2)x + c = 0

**Step 4 (Simplify):** Take the common factors out and simplify.

### Example 6.4.2:

Factorise the following expressions:**(a) **\( \scriptsize 5x^2 \: – \: 7x \: + \: 2\)**(b) **\( \scriptsize 35x^2 \: + \: 31x\: + \: 6\)**(c)** \( \scriptsize 20x^2\: – \: 40xy \:+ \:15y^2\)**(d)** \( \scriptsize 8x^3 \:- \:44x^2 \:+ \:20x\)

**Solution**

**(a)** \( \scriptsize 5x^2\: – 7x \:+\: 2\)

**Step 1:** Multiply the coefficient of **x ^{2}** and the constant term

**c**i.e.

**a**×

**c**=

**ac**

a = 5, c = 2

∴ 5 × 2 = 10

**Step 2:** Find two numbers such that their product is equal to ac and sum equals to b.

**Product **= \( \scriptsize -5x \: \times \: -2x = \: -10x^2 \)

**Sum **= \( \scriptsize -5x \: + \: -2x = \: -7x\)

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