The below examples show that the method of factorizing ax^{2} + bx + c by breaking the middle term involves the following steps.
Steps:
1.Take the product of the constant term and the coefficient of x^{2}, i.e., ac.
2. Break ac into two factors p, q whose sum is b, i.e., p + q = b.
3. Pair one of them, say px, with ax^2 and the other, qx, with c. Then factorize the expression.
Solved Examples on Factorization of Expressions of the Form ax^2 + bx + c, a ≠ 1:
1. Factorize: 6m^{2} + 7m + 2.
Solution:
Here, 6 × 2 = 12 = 3 × 4 and, 3 + 4 = 7 (= coefficient of m).
Therefore, 6m^{2} + 7m + 2 = 6m^{2} + 3m + 4m + 2
= 3m(2m + 1) + 2(2m + 1)
= (2m + 1)(3m + 2)
2. Factorize: 1 – 18x – 63x^{2}
Solution:
The given expression is – 63x^{2} - 18x + 1
Here, (-63) × 1 = -63 = (-21) × (3), and -21 + 3 = -18(= coefficient of x).
Therefore, – 63x^{2} - 18x + 1 = – 63x^{2} – 21x + 3x + 1
= -21x(3x + 1) + 1(3x + 1)
= (3x + 1)(-21x + 1)
= (1 + 3x)(1 – 21x).
3. Factorize: 6x^{2} – 7x – 5.
Solution:
6 × (-5) = -30 = (-10) × (3), and -10 + 3 = - 7 (= coefficient of x).
Therefore, 6x^{2} – 7x – 5 = 6x^{2} – 10x + 3x – 5
= 2x(3x – 5) + 1(3x – 5)
= (3x – 5)(2x + 1)
4. Factorize: 30m^{2} + 103mn – 7n^{2}
Solution:
30 × (-7) = -210 = (105) × (-2), and 105 + (-2) = 103 (= coefficient of mn).
Therefore the given expression, 30m^{2} + 103mn – 7n^{2}
= 30m^{2} + 105mn – 2mn – 7n^{2}
= 15m(2m + 7n) – n(2m + 7n)
= (2m + 7n)(15m – n)
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