Practice the questions given in the worksheet on factorization of the trinomial ax^{2} + bx + c.
1. Factorization of a perfect-square trinomial.
(i) a^{2} + 6a + 9
(ii) a^{2} + a + \(\frac{1}{4}\)
(iii) 25x^{2} – 10x + 1
(iv) 4x^{2} – 4xy + y^{2}
2. Factorization of expressions of the form x^{2} + (a + b)x + ab
(i) x^{2} + 12x + 35
(ii) a^{2} + 13a + 42
(iii) x^{2} + 15x + 50
(iv) x^{2} + 4x + \(\frac{15}{4}\)
(v) 18 + 11x + x^{2 }
(vi) a^{2} + 3ab + 2b^{2}
3. Factorize:
(i) x^{2} – 15x + 50
(ii) a^{2} – 13a + 42
(iii) p^{2} – 5p + \(\frac{9}{4}\)
(iv) x^{2} – 7xy + 12y^{2}
(v) 21 – 10m + m^{2}
(vi) 51 – 20x + x^{2}
4. Factorize:
(i) a^{2} – a – 42
(ii) x^{2} + x – 56
(iii) x^{2} – 5x – 84
(iv) x^{2} + 5x – 84
(v) x^{2} – xy – 72y^{2}
(vi) m^{2} + 2m - \(\frac{5}{4}\)
5. Factorization of expressions of the form ax^{2} + bx + c, a ≠ 1.
(i) 10a^{2} + 17a + 3
(ii) 5x^{2} + 6x + 1
(iii) 6x^{2} – 17x + 12
(iv) 2x^{2} – x – 6
(v) 8a^{2} – 21a + 10
(vi) 7 – 4a – 3a^{2}
(vii) 3x^2 – 1x + 6
(viii) 20x^{2} – x – 1
(ix) 12x^{2} – 4x – 5
(x) 2x^{2} + 5xy + 2y^{2}
6. Miscellaneous factorizations:
(i) (x + y)^{2} – 3x – 3y + 2
[Hint: Given expression = a^{2} – 3a + 2, where a = x + y
= (a – 1)(a – 2)
= (x + y – 1)(x + y – 2).]
(ii) (x + 1)^{2} + x – 5
(iii) (x + 2)(x + 3) – 12
(iv) (x – 1)(x + 4) – 50
(v) (x + 1)^{2} + (x + 2)^{2} – 13
(vi) x^{2} – (a + \(\frac{1}{a}\))x + 1
[Hint: Given expression = x^{2} – ax - \(\frac{1}{a}\)x + a ∙ \(\frac{1}{a}\) = x(x – a) - \(\frac{1}{a}\)(x - a) = (x – a)(x - \(\frac{1}{a}\)).]
(vii) 2a^{3}x^{2} – 5a^{2}x – 12a.
[Hint: Given expression = a(2a^{2}x^{2} – 5ax – 12) = a(2y^{2} – 5y – 12), where ax = y.]
7. Application on factorization:
(i) x^{2} + 2x – 15 and 3x^{2} – 11x + 6
(ii) x^{2} – x – 2 and 6x^{2} + x – 5.
Answers:
1. (i) (a + 3)(a + 3)
(ii) (a + \(\frac{1}{2}\))(a + \(\frac{1}{2}\))
(iii) (5x – 1)(5x – 1)
(iv) (2x – y)(2x – y)
2. (i) (x + 7)(x + 5)
(ii) (a + 6)(a + 7)
(iii) (x + 5)(x + 10)
(iv) (x + \(\frac{5}{2}\))(x + \(\frac{3}{2}\))
(v) (x + 9)(x + 2)
(vi) (a + b)(a + 2b)
3. (i) (x – 10)(x – 5)
(ii) (a – 6)(a – 7)
(iii) (p - \(\frac{9}{2}\))(p - \(\frac{1}{2}\))
(iv) (x – 4y)(x – 3y)
(v) (m – 7)(m – 3)
(vi) (x – 3)(x – 17)
4. (i) (a – 7)(a + 6)
(ii) (x + 8)(x – 7)
(iii) (x – 12)(x + 7)
(iv) (x + 12)(x – 7)
(v) (x – 9y)(x + 8y)
(vi) (m + \(\frac{5}{2}\))(m - \(\frac{1}{2}\))
5. (i) (5a + 1)(2a + 3)
(ii) (5x + 1)(x + 1)
(iii) (2x – 3)(3x – 4)
(iv) (x – 2)(2x + 3)
(v) (a – 2)(8a – 5)
(vi) (7 + 3a)(1 – a)
(vii) (3x – 2)(x – 3)
(viii) (5x + 1)(4x – 1)
(ix) (6x – 5)(2x + 1)
(x) (2x + y)(x + 2y)
6. (i) (x + y – 1)(x + y – 2)
(ii) (x + 4)(x – 1)
(iii) (x + 6)(x – 1)
(iv) (x + 9)(x – 6)
(v) 2(x + 4)(x – 1)
(vi) (x – a)(x - \(\frac{1}{a}\))
(vii) 2a^3x^2 – 5a^2x – 12a.
7. (i) LCM = (x – 3)(x + 5)(3x – 2), HCF = x - 3
(ii) LCM = (x – 2)(x + 1)(6x – 5), HCF = x + 1
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