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Question 512 Marks

Factorise: $16 x^2-24 x+9$

Answer

$16 x^2-24 x+9$
$=(4 x)^2-2 \times 4 x \times 3+(3)^2$
$=(4 x-3)^2$

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Question 522 Marks

Factorise: $ab^2+ (a - 1)b - 1$

Answer

$= ab^2+ ab - b - 1$
$= ab(b + 1) - 1(b + 1)$
$= (b + 1)(ab - 1)$

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Question 532 Marks

Evalute: ${(7.8)^2- (2.2)^2}$

Answer

$(7.8)^2- (2.2)^2$
$= (7.8 + 2.2)(7.8 - 2.2)$
$= 10.00 × 5.6$
$= 56$

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Question 542 Marks

Factorise: $x^2- xz + xy - yz$

Answer

$(x-2 y)^2+4 x-8 y$
$=(x-2 y)^2+4(x-2 y)$
$=(x-2 y)(x-2 y+4)$

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Question 552 Marks

Factorise: $a^2+6 a-91$

Answer

$a^2+6 a-91$
$=a^2+13 a-7 a-91$
${-91 = 13 × (-7), 6 = 13 - 7}$
$= a(a + 13) - 7(a + 13)$
$ = (a + 13)(a - 7)$

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Question 562 Marks

Factorise: $x^2+13 x+40$

Answer

$x^2+13 x+40$
$=x^2+5 x+8 x+40$
${40 = 5 × 8, 13 = 5 + 8}$
$= x(x + 5) + 8(y + 5) $
$= (x + 5)(x + 8)$

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Question 572 Marks

Factorise: $x^2-22 x+117$

Answer

$x^2-22 x+117$
$=x^2-13 x-9 x+117$
${117 = (-13) × (-9), -22 = -13 - 9}$
$= x(x - 13) - 9(y - 13) $
$= (x - 9)(x - 13)$

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Question 582 Marks

Factorise: $m^2-4 m n+4 n^2$

Answer

$m^2-4 m n+4 n^2$
$=(m)^2-2 \times m \times 2 n+(2 n)^2$
$=(m-2 n)^2$

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Question 592 Marks

Factorise: $4 n^2-8 n+3$

Answer

$4 n^2-8 n+3$
$=4 n^2-6 n-2 n+3$
${4 × 3 = 12, 12 = (-6) × (-2), - 8 = -6 - 2}$
$= 2n(2n - 3) - 1(2n - 3)$
$ = (2n - 3)(2n - 1)$

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Question 602 Marks

Factorise: $(l+m)^2-41 m$

Answer

$(l+m)^2-41 m$
$=l^2+m^2+21 m-41 m$
$=l^2+m^2-21 m=l^2-21 m+m^2$
$=(l-m)^2$
$\{\because a^2+ 2ab + b^2= (a + b)^2\}$

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Question 612 Marks

Factorise: $9+6 z+z^2$

Answer

$9+6 z+z^2$
$=(3)^2+2 \times 3 \times z+(z)^2 $
$ =(3+z)^2 $

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Question 622 Marks

Factorise: $z^2-12 z-45$

Answer

$z^2-12 z-45$
$=z^2-15 z+3 z-45$
${-45 = -15 × 3, -12 = -15 + 3}$
$= z(z - 15) + 3(z - 15) $
$= (z - 15)(z + 3)$

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Question 632 Marks

Factorise:
$=y^2-x y(1-x)-x^3$

Answer

$=y^2-x y(1-x)-x^3$
$=y^2-x y+x^2 y-x^2$
$=y(y-x)+x^2(y-x)$
$=(y-x)\left(y+x^2\right)$

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Question 642 Marks

Factorise: $y^2-6 y-135$

Answer

$y^2-6 y-135$
$=y^2-15+9 y-135$
${-135 = -15 × 9, -6 = -15 + 9}$
$= y(y - 15) + 9(y - 15)$
$= (y - 15)(y + 9)$

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Question 652 Marks

Factorise: $3 z^2-10 z+8$

Answer

$3 z^2-10 z+8$
$=3 z^2-6 z-4 z+8$
$\{24 = 3 × 8, 24 = (-6) × (-4), -10 = -6 - 4\}$
$= 3z(z - 2) - 4(z - 2) $
$= (z - 2)(3z - 4)$

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Question 662 Marks

Factorise: $ 9 y^2-12 y+4$

Answer

$ 9 y^2-12 y+4$
$=(3 y)^2-2 \times 3 y \times 2+(2)^2 $
$\{\because a 2-2 a b+b 2=(a-b) 2\}$
$ =(3 y-2)^2$

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Question 672 Marks

Factorise: $3 y^2+14 y+8$

Answer

$3 y^2+14 y+8$
$=3 y^2+12 y+2 y+8$
${24 = 3 × 8, 24 = 12 × 2, 14 = 12 + 2}$
$= 3y(y + 4) + 2(y + 4) $
$= (y + 4)(3y + 2)$

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Question 682 Marks

Factorise: $a b^2-b c^2-a b+c^2 $

Answer

$a b^2-b c^2-a b+c^2 $
$a b^2-a b-b c^2+c^2$
$=a b(b-1)-c^2(b-1)$
$=(b-1)\left(a b-c^2\right)$

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Question 692 Marks

Factorise: $(x+5)^2-4(x+5)$

Answer

$(x+5)^2-4(x+5)$
$= (x + 5)(x + 5 - 4) $
$= (x + 5)(x + 1)$

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Question 702 Marks

Factorise: $x^2- ax - bx + ab$

Answer

$x^2- ax - bx + ab$
$= x(x - a) - b(x - a)$
$ = (x - a)(x - b)$

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Question 712 Marks

Factorise: $36 a^2+36 a+9$

Answer

$36 a^2+36 a+9$
$=9\left[4 a^2+4 a+1\right]$
$=9\left[(2 a)^2+2 \times 2 a x+(1)^2\right]$
$=9[2 a+1]^2$

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Question 722 Marks

Factorise: $p^2-10 p+25$

Answer

$p^2-10 p+25$
$=(p)^2-2 \times p \times 5+(5)^2$
$=(p-5)^2$
$\{\because a^2- 2ab + b^2= (a - b)^2\}$

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Question 732 Marks

Factorise: $3(a-2 b)^2-5(a-2 b)$

Answer

$3(a-2 b)^2-5(a-2 b)$
$ = (a - 2b)\{3(a - 2b) - 5\}$
$= (a - 2b)(3a - 6b - 5)$

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Question 742 Marks

$36c^2- (5a + b)^2$

Answer

$= (6c)^2- (5a + b)^2$
$[\because a^2- b^2= (a + b)(a - b)\}$
$= (6c + 5a + b)(6c - 5a - b)$

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Question 752 Marks

Factorise: $12(2 x-3 y)^2-16(3 y-2 x)$

Answer

$12(2 x-3 y)^2-16(3 y-2 x)$
$=12(2 x-3 y)^2+16(2 x-3 y)$
$ = 4(2x - 3y){3(2x - 3y) + 4} $
$= 4(2x - 3y)(6x - 9y + 4)$

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Question 762 Marks

Factorise:
$9a(3a - 5b) - 12a^2(3a - 5b)$

Answer

$9a(3a - 5b) - 12a^2(3a - 5b) $
$= 3a(3a - 5b)(3 - 4a)$

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Question 772 Marks

Factorise:
$ x^2+14 x+49$

Answer

$ x^2+14 x+49$
$=(x)^2+2 \times x \times 7+(7)^2 $
$ =(x+7)^2 $

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Question 782 Marks

Factorise:
$x^2+15 x+56$

Answer

$x^2+15 x+56$
$=x^2+8 x+7 x+56$
${56 = 8 × 7, 15 = 8 + 7} $
$= x(x + 8) + 7(x + 8) $
$= (x + 8)(x + 7)$

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Question 792 Marks

Factorise: $\text{z}^2+\text{z}+\frac{1}{4}$

Answer

$\text{z}^2+\text{z}+\frac{1}{4}$ $=(\text{z})^2+2\times\text{z}\times\frac{1}{2}+\Big(\frac{1}{2}\Big)^2$ $=\Big(\text{z}+\frac{1}{2}\Big)^2$

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Question 802 Marks

Factorise: $16 p^3-4 p$

Answer

$16 p^3-4 p$
$=4 p\left[4 p^2-1\right]$
$=4 p(2 p)^2-(1)^2$
$=4 p(2 p+1)(2 p-1)$

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Question 812 Marks

Factorise: $y^2+7 y-144$

Answer

$y^2+7 y-144$
$=y^2+16 y-9 y-144$
${144 = -16 × 9, 7 = 16 - 9}$
$= y(y + 16) - 9(y + 16) $
$= (y + 16)(y - 9)$

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Question 822 Marks

Factorise:
$16 a^2-144$

Answer

Using: $a^2-b^2=(a+b)(a-b) $
$16 a^2-144=(4 a)^2=(12)^2$
$=(4 a+12)(4 a-12)$
$=4(a+3) \times 4(a-3)$
$=16(a+3)(a-3)$

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Question 832 Marks

Factorise: $3+23 z-8 z^2$

Answer

$3+23 z-8 z^2$
$=3+24 z-z-8 z^2$
$\{3 × (-8) = -24, -24 = 24 × (-1), 23 = 24 - 1\}$
$= 3(1 + 8z) - z(1 + 8z)$
$ = (1 + 8z)(3 - z)$

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Question 842 Marks

Factorise:
$9 m^2+24 m+16$

Answer

$9 m^2+24 m+16$
$=9(3 m)^2+2 \times 3 m \times 4+(4)^2] $
$ =(3 m+4)^2 $
$\{\therefore a2 + 2ab + b2 = (a + b)2\}$

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Question 852 Marks

Factorise:
$25 a^2-4 b^2+28 b c-49 c^2$

Answer

$25 a^2-4 b^2+28 b c-49 c^2$
$=25-\left[4 b^2-28 b c+49 c^2\right]$
${\left[\because a^2-2 a b+b^2=(a-b)^2\right]}$
$=(5 a)^2-\left[(2 b)^2-2 \times 2 b \times 7 c+(7 c)^2\right] $
$ =(5 a)^2-(2 b-7 c)^2 $
$\left\{\because\left(a^2-b^2=(a+b)(a-b)\right\}\right.$
$ = (5a + 2b - 7c)(5a - 2b + 7c)$

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Question 862 Marks

Factorise: $ x^2-36 $

Answer

$ x^2-36 $
$ =(x)^2-(6)^2\left\{\therefore a^2-b^2=(a+b)(a-b)\right\} $
$ =(x+6)(x-6) $

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Question 872 Marks

Factorise:
$x^2-4 x-12$

Answer

$x^2-4 x-12$
$=x^2-6 x+2 x-12$
$\{-12 = -6 × 2, -4 = -6 + 2\}$
$= x(x - 6) + 2(x - 6) $
$= (x - 6)(x + 2)$

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Question 882 Marks

Factorise:
$ (a x+b y)^2+(b x-a y)^2$

Answer

$ (a x+b y)^2+(b x-a y)^2$
$= (a^2 x^2+b^2 y^2+2 a x b y+b^2 x^2+a^2 y^2-2 b x a y$
$=a^2 x^2+b^2 y^2+b^2 x^2+a^2 y^2$
$=a^2 x^2+b^2 x^2+a^2 y^2+b^2 y^2$
$= x^2\left(a^2+b^2\right)+y^2\left(a^2+b^2\right)$
$=\left(a^2+b^2\right)\left(x^2+y^2\right) $

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Question 892 Marks

$x^2-y^2-2 y-1$

Answer

$x^2-y^2-2 y-1$
$=x^2-\left(y^2+2 y+1\right)$
$=(x)^2-(y+1)^2$
$=(x+y+1)(x-y-1)$

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Question 902 Marks

Factorise:
$25-a^2-b^2-2 a b$

Answer

$25-a^2-b^2-2 a b$
$=25-\left(a^2+b^2+2 a b\right)$
$=(5)^2-(a+b)^2$
$=(5+a+b)(5-a-b)$

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Question 912 Marks