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SS1: MATHEMATICS - 2ND TERM

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  • Factorization of Quadratic Expressions II
    • Quadratic Expression of the form ax2 + bx + c

Quadratic Expression of the form ax2 + bx + c

Factorization of Quadratic Equation by Splitting the Middle Term:

In order to factorise a quadratic expression of the form ax2 + bx + c, where \( \scriptsize a \neq 1 \)  

Consider the quadratic equation ax2 + bx + c = 0

Step 1: Multiply the coefficient of x2 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,

x2 + (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 x2 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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