Factorization of Quadratic Trinomials
In factorization of quadratic trinomials there are two forms:
(i) First Form
(ii) Second Form
(i) First Form of Factorization of quadratic trinomials
x2 + px +q
Suppose we are given a quadratic trinomial x2 + px + q. Then, we use the identity:
x2 + (a + b) × + ab = (x + a)(x + b).
In order to factorize x2 + px + q, we find two numbers a and b such that (a + b) = p and ab = q.
Then,
x2 + px + q = x2 + (a + b) x + ab = (x + a)(x + b).
1. Factorize:
(i) x2 + 8x + 15
Solution:
The given expression is x2 + 8x + 15.
Find two numbers whose sum = 8 and product = 15.
Clearly, the numbers are 5 and 3.
Therefore, x2 + 8x + 15 = x2 + 5x + 3x + 15
= x (x + 5) + 3 (x + 5).
= (x + 5)(x + 3).
(ii) x2 + 15x + 56
Solution:
The given expression is x2 + 15x + 56.
Find two numbers whose sum = 15 and product = 56.
Clearly, such numbers are 8 and 7.
Therefore, x2 + 15x + 56 = x2 + 8x + 7x + 56
= x (x + 8) + 7(x + 8)
= (x + 8)(x + 7).
2. Factorize:
(i) x2 - 7x + 12
Solution:
The given expression is x2 - 7x + 12
Find two numbers whose sum = -7 and product = 12
Clearly, such numbers are (-4) and (-3).
Therefore, x2 - 7x + 12 = x2 - 4x - 3x + 12
= x (x - 4) -3 (x - 4)
= (x - 4)(x - 3).
(ii) x2 + x - 56
Solution:
The given expression is x2 + x - 56.
Find two numbers whose sum = 1 and product = -56.
Clearly, such numbers are 8 and -7.
Therefore, x2 + x - 56 = x2 + 8x - 7x - 56
= x (x + 8) -7 (x + 8)
= (x + 8)(x - 7).
Second Form of Factorization of quadratic trinomials
ax2 + bx + c
In this case, we split b into two parts whose sum = b and product = ac. Now, we proceed as in the first case.
1. Factorize:
(i) 2x2 + 9x + 10
Solution:
The given expression is 2x2 + 9x + 10.
Find two numbers whose sum = 9 and product = (2 × 10) = 20.
vClearly, 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:
(i) 15x2 - 26x + 8
Solution:
The given expression is 15x2 - 26x + 8.
Find two numbers whose sum = -26 and product = (15 × 8) = 120.
Clearly, such numbers are -20 and -6.
Therefore, 15x2 - 26x + 8 = 15x2 - 20x - 6x + 8
= 5x(3x - 4) -2 (3x - 4)
= (3x - 4)(5x - 2).
(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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