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 $2x^2+5x=a$

$= (a)(a-19)+84$

$= a^2-19a+84$

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

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

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

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

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

$\\$

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) $7x^2+2\sqrt{14}x+2$

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

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

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

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

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

$\\$