First half of Exercise 6(a)…

Question 7: Factorize:

(i) x^2+12x-45

= x^2+15x-3x-45

= x(x+15)-3(x+15)

= (x+15)(x-3)

(ii) x^2-22x+120

= x^2 - 12x - 10x + 120

= x(x-12)-10(x-12)

= (x-10)(x-12)

(iii) x^2-11x-42

= x^2-14x+3x-42

= x(x-14)+3(x-14)

= (x+3)(x-14)

(iv) y^2+5y-36

= y^2 +9y-4y-36

= y(y+9)-4(y+9)

=(y-4)(y+9)

(v) (a+b)^2-5(a+b)+4

Let a+b = x

= x^2-5x+4

= x^2-4x-x+ 4

= x(x-4)-1(x-4)

= (x-4)(x-1)

= (a+b-4)(a+b-4)

(vi) 3(x+y)^2-5(x+y)+2

Let x+y = a

= 3a^2-5a+2

= 3a^2 - 3a - 2a + 2

= 3a(a-1)-2(a-1)

=(a-1)(3a-2)

= (x+y-1)(3x+3y-2)

(vii) (p^2+4p)^2+21(p^2+4p)+98

Let p^2+4p = x

= x^2 + 21x + 98

= x^2 + 14x + 7x + 98

= x(x+14) + 7( x + 14)

= (x+14)(x+7)

= (p^2+4p+14)(p^2+4p+7)

(viii) x^2-\sqrt{3}x-6

= x^2 -2\sqrt{3}x+\sqrt{3}x-6

= x(x+\sqrt{3})-2\sqrt{3}(x+\sqrt{3})

= (x+\sqrt{3})(x-2\sqrt{3})

(ix) x^2+5\sqrt{5}x+30

= x^2 + 3\sqrt{5} x + 2\sqrt{5} x + 3\sqrt{5} \times 2\sqrt{5}

= x(x+3\sqrt{5}) + 2\sqrt{5}(x+3\sqrt{5})

= (x+3\sqrt{5})(x+2\sqrt{3})

(x) (2x^2+5x)(2x^2+5x-19)+84

Let 2x^2+5x=a

= (a)(a-19)+84

= a^2-19a+84

= a^2-12a-7a+84

= a(a-12)-7(a-12)

= (a-12)(a-7)

= (2x^2+5x-12)(2x^2+5x-7)

= (2x-3)(x+4)(2x+7)(x-1)

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Question 8: Factorize

(i) 5x^2-32x+12

= 5x^2-30x-2x+12

= 5x(x-6)-2(x-6)

= (x-6)(5x-2)

(ii) 30x^2+7x-15

= 30x^2+25x-18x-15

= 5x(6x+5)-3(6x+5)

=(6x+5)(5x-3)

(iii) 6x^2-\sqrt{5}x-5

= 6x^2-3\sqrt{5}x+2\sqrt{5}x-5

= 3x(2x-\sqrt{5})+\sqrt{5}(2x-\sqrt{5})

= (2x-\sqrt{5})(3x+\sqrt{5})

(iv) \frac{1}{2} x^2-3x+4

= \frac{1}{2} x^2-2x-x+4

= \frac{1}{2} x(x-4)-1(x-4)

= (x-4)( \frac{1}{2} x -1)

(v) 4\sqrt{3}x^2+5x-2\sqrt{3}

= 4\sqrt{3}x^2 + 8x - 3x - 2\sqrt{3}

=4x(\sqrt{3}x+2) - \sqrt{3}(\sqrt{3}x+2)

= (\sqrt{3}x+2)(4x-\sqrt{3})

(vi) \frac{a}{b} x^2+( \frac{a}{b} + \frac{c}{d}) x + \frac{c}{d}

= \frac{a}{b} x (x+1) + \frac{c}{d} (x+1)

= (x+1) ( \frac{a}{b} x+ \frac{c}{d} )

(vii) px^2+(4p^2-3q)x-12pq

= px^2 + 4p^2x-3qx-12pq

= px(x+4p)-3q(x+4p)

= (x+4p)(px-3q)

(viii) 3(a-2)^2-2(a-2)-8

Let a-2 = x

= 3x^2-2x-8

= 3x^2 - 6x+4x-8

= 3x(x-2)+4(x-2)

= (x-2)(3x+4)

(ix) 12(a+1)^2-25(a+1)(b+2)+12(b+2)^2

Let a+1 = x and b+2 = y

= 12x^2 - 25xy + 12y^2

= 12x^2 - 9xy - 16xy + 12y^2

= 3x(4x-3y) + 4y(4x-3y)

= (4x-3y)(3x+4y)

= \Big( 4(a+1)-3(b+2) \Big) \Big( 3(a+1)+4(b+2) \Big)

= (4a-3b-2)(3a-4b-5)

(x) 5x^6-7x^3-6

Let x^3 = a

= 5a^2-7a-6

= 5a^2-10a+3a-6

= 5a^2(a-2)+3(a-2)

= (a-2)(5a+3)

=(x^3-2)(5x^3+3)

(xi) x^2+ \frac{12}{35} x+ \frac{1}{35}

= x^2 + \frac{1}{7} x + \frac{1}{5} x + \frac{1}{35}

= x(x+ \frac{1}{7} )+ \frac{1}{5} (x+ \frac{1}{7} )

= (x+ \frac{1}{7} )(x+ \frac{1}{5} )

(xii) 2x^2+3\sqrt{3}x+3

= 2x^2 + 2\sqrt{3} x + \sqrt{3} x + 3

= 2x(x+\sqrt{3}) + \sqrt{3}(x+\sqrt{3})

= (x+\sqrt{3})(2x+\sqrt{3})

(xiii) 5\sqrt{5}x^2+20x+3\sqrt{5}

= 5\sqrt{5} x^2 + 15x + 5 x + 3\sqrt{5}

= \sqrt{5}x(5x+\sqrt{5}) + 3 (5x+\sqrt{5})

= (5x+\sqrt{5})(\sqrt{5}x+3)

(xiv) 2x^2+3\sqrt{5}x+5

= 2x^2+2\sqrt{5}x+\sqrt{5}x+5

= 2x(x+\sqrt{5}) + \sqrt{5}(x+\sqrt{5})

= (x+\sqrt{5})(2x+\sqrt{5})

(xv) 7x^2+2\sqrt{14}x+2

= 7x^2+\sqrt{14} x + \sqrt{14} x + 2

= 7x^2+\sqrt{2 \times 7} x + \sqrt{2 \times 7} x + 2

= \sqrt{7}x(\sqrt{7}x+\sqrt{2})+\sqrt{2}(\sqrt{7}x+\sqrt{2})

= (\sqrt{7}x + \sqrt{2})(\sqrt{7}x+\sqrt{2})

= (\sqrt{7}x + \sqrt{2})^2

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