Samacheer Kalvi 9th Maths Solutions Chapter 3 Algebra Ex 3.6

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Tamilnadu Samacheer Kalvi 9th Maths Solutions Chapter 3 Algebra Ex 3.6

Question 1.
Factorise the following:
(i) x² + 10x + 24
(ii) z² + 4z – 12
(iii) p² – 6p – 16
(iv) t² + 72 – 17t
(v) y² – 16y – 80
(vi) a² + 10a – 600
Solution:
(i) x² + 10x + 24
x² + 10x + 24 = x² + 6x + 4x + 24

= x(x + 6) + 4 (x + 6)
= (x + 6) (x + 4)

(ii) z² + 4z – 12
z² + 4z – 12 = z² + 6z – 2z- 12

= z (z + 6) – 2 (z + 6)
= (z + 6) (z – 2)

(iii) p² – 6p – 16
p² – 6p – 16 = p² – 8p + 2p – 16

= p(p – 8) + 2(p – 8)
= (p – 8)(p + 2)

(iv) t² + 72 – 17t
t² + 72 – 17t = t² – 17t + 72
= t² – 9t – 8t + 72

= t(t – 9) – 8 (t – 9)
= (t – 9) (t – 8)

(v) y² – 16y – 80
y² – 16y – 80 = y² – 20y + 4y – 80

= y(y – 20) + 4 (y – 20)
= (y – 20) (y + 4)

(vi) a² + 10a – 600
a² + 10a – 600 = a² + 30a – 20a – 600

= a(a + 30) -20 (a + 30)
= (a + 30) (a – 20)

Question 2.
Factorise the following
(i) 2a² + 9a + 10
(ii) 5x² – 29xy – 42y²
(iii) 9 – 18x + 18x²
(iv) 6x² + 16xy + 8y²
(v) 12x² + 36x²y + 27y²x²
(vi) (a + b)² + 9 (a + b) + 18
Solution:
(i) 2a² + 9a + 10
2a² + 9a + 10 = 2a² + 4a + 5a + 10

= 2a(a + 2) + 5 (a + 2)
= (a+ 2) (2a+ 5)

(ii) 5x² – 29xy – 42y²
5x² – 35xy + 6xy – 42y²

= 5x (x – 7) + 6y (x – 7)
= (x – 7) (5x + 6y)

(iii) 9 – 18x + 8x²
= 8x² – 18x + 9
= 8x² – 6x – 12x + 9

= 2x (4 x – 3) – 3 (4x – 3)
= (4x – 3) (2x – 3)

(iv) 6x² + 16xy + 8y²
= 2 (3x² + 8xy + 4y²)
= 2 (3x² + 8xy + 4y²)

= 2 (3x² + 6xy + 2xy + 4y²)
= 2 (3x (x + 2y) + 2y (x + 2y))
= 2 (x + 2y) (3x + 2y)

(v) 12x² + 36x²y + 27y²x²
= 27y²x² + 36x²y + 12x² = 3x²(9y² + 12y + 4)

= 3x² (9y² + 6y + 6y + 4) = 3x² (3y (3y + 2) + 2 (3y + 2))
= 3x² (3y + 2) (3y + 2) = 3x² (3y + 2) (3y + 2)

(vi) (a + b)² + 9 (a + 6) + 18
= (a + b)² + 6 (a + b) + 3 (a + b) + 18

= (a + b) ((a + b) + 6) + 3 ((a + b) + 6)
= ((a + 6) + 6) ((a + b) + 3) = (a + b + 6) (a + b + 3)

Question 3.
Factorise the following:
(i) (p – q)² – 6(p – q) – 16
(ii) m² + 2mn – 24n²
(iii) \(\sqrt{5} a^{2}\) + 2a – \(3 \sqrt{5}\)
(iv) a⁴ – 3a² + 2
(v) 8m³ – 2m²n – 15mn²
(vi) \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{2}{x y}\)
Solution:
(i) (p – q)² – 6 (p – q) – 16
= (p – q)² – 8(p – q) + 2(p – q) – 16

= (p – q)((p – q) – 8) + 2((p – q) – 8)
= (p – q – 8)(p – q + 2)

(ii) m² + 2mn – 24n²
= m² + 6mn – 4mn – 24n²

= m(m + 6n) – 4n(m + 6n)
= (m + 6n)(m – 4n)