Here we will solve different types of Problems on Factorization of Expressions of the Form x^{2} + (a + b)x + ab.
1. Factorize: a^{2} + 25a - 54
Solution:
Here, constant term = -54 = (27) × (-2), and 27 + (-2) = 25 (= coefficient of a).
Therefore, a^{2} + 25a – 54 = a^{2} + 27a - 2a - 54 (breaking 25a is sum of two terms, 27a - 2a)
= (a^{2} + 27a) + (- 2a - 54)
= a(a + 27) - 2(a + 27)
= (a + 27)(a - 2).
2. Factorize: 3 - 4p + p^{2}
Solution:
Here, constant term = 3 = (-3) × (-1), and (-3) + (-1) = -4 (= coefficient of p).
Therefore, 3 - 4p + p^{2} = p^{2} – 4p + 3
=p^{2 }– 3p – p + 3 (breaking -4p is sum of two terms, -3p - p)
= (p^{2} – 3p) + (- p + 3)
= p(p - 3) - 1(p - 3)
= (p - 3)(p - 1).
3. Factorize: x^{2} – xy – 30y^{2}
Solution:
Here, -30 = (-6) × 5, and (-6) + 5 = -1 (= coefficient of xy).
Therefore, x^{2} – xy – 30y^{2} = x^{2} – 6xy + 5xy – 30y^{2} (breaking -xy is sum of two terms, -6xy + 5xy)
= (x^{2} – 6xy) + (5xy – 30y^{2})
= x(x – 6y) + 5y(x – 6y)
= (x – 6y)(x + 5y).
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