Problems on Factorization of Expressions of the Form
x$^{2}$ +(a + b)x +ab

Here we will solve different types of Problems on Factorization of Expressions of the Form x² + (a + b)x + ab.

1. Factorize: a² + 25a - 54

Solution:

Here, constant term = -54 = (27) × (-2), and 27 + (-2) = 25 (= coefficient of a).

Therefore, a² + 25a – 54 = a² + 27a - 2a - 54 (breaking 25a is sum of two terms, 27a - 2a)

                                    = (a² + 27a) + (- 2a - 54)

                                    = a(a + 27) - 2(a + 27)

                                    = (a + 27)(a  - 2).

2. Factorize: 3 - 4p + p²

Solution:

Here, constant term = 3 = (-3) × (-1), and (-3) + (-1) = -4 (= coefficient of p).

Therefore, 3 - 4p + p² = p² – 4p + 3

                                  =p² – 3p – p + 3 (breaking -4p is sum of two terms, -3p - p)

                                  = (p² – 3p) + (- p + 3)

                                  = p(p - 3) - 1(p - 3)

                                  = (p - 3)(p - 1).

3. Factorize: x² – xy – 30y²

Solution:

Here, -30 = (-6) × 5,  and (-6) + 5 = -1 (= coefficient of xy).

Therefore, x² – xy – 30y² = x² – 6xy + 5xy – 30y² (breaking -xy is sum of two terms, -6xy + 5xy)

                                      = (x² – 6xy) + (5xy – 30y²)

                                      = x(x – 6y) + 5y(x – 6y)

                                      = (x – 6y)(x + 5y).

9th Grade Math

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