Worksheet on Factorization of the Trinomial ax\(^{2}\) + bx + c

Practice the questions given in the worksheet on factorization of the trinomial ax² + bx + c.

1. Factorization of a perfect-square trinomial.

(i) a² + 6a + 9

(ii) a² + a + \(\frac{1}{4}\)

(iii) 25x² – 10x + 1

(iv) 4x² – 4xy + y²

2. Factorization of expressions of the form x² + (a + b)x + ab

(i) x² + 12x + 35

(ii) a² + 13a + 42

(iii) x² + 15x + 50

(iv) x² + 4x + \(\frac{15}{4}\)

(v) 18 + 11x + x²

(vi) a² + 3ab + 2b²

3. Factorize:

(i) x² – 15x + 50

(ii) a² – 13a + 42

(iii) p² – 5p + \(\frac{9}{4}\)

(iv) x² – 7xy + 12y²

(v) 21 – 10m + m²

(vi) 51 – 20x + x²

4. Factorize:

(i) a² – a – 42

(ii) x² + x – 56

(iii) x² – 5x – 84

(iv) x² + 5x – 84

(v) x² – xy – 72y²

(vi) m² + 2m - \(\frac{5}{4}\)

5. Factorization of expressions of the form ax² + bx + c, a ≠ 1.

(i) 10a² + 17a + 3

(ii) 5x² + 6x + 1

(iii) 6x² – 17x + 12

(iv) 2x² – x – 6

(v) 8a² – 21a + 10

(vi) 7 – 4a – 3a²

(vii) 3x^2 – 1x + 6

(viii) 20x² – x – 1

(ix) 12x² – 4x – 5

(x) 2x² + 5xy + 2y²

6. Miscellaneous factorizations:

(i) (x + y)² – 3x – 3y + 2

[Hint: Given expression = a² – 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) x² – (a + \(\frac{1}{a}\))x + 1

[Hint: Given expression = x² – ax - \(\frac{1}{a}\)x + a ∙ \(\frac{1}{a}\) = x(x – a) - \(\frac{1}{a}\)(x - a) = (x – a)(x - \(\frac{1}{a}\)).]

(vii) 2a³x² – 5a²x – 12a.

[Hint: Given expression = a(2a²x² – 5ax – 12) = a(2y² – 5y – 12), where ax = y.]

 

7. Application on factorization:

(i) x² + 2x – 15 and 3x² – 11x + 6

(ii) x² – x – 2 and 6x² + 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

9th Grade Math

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