Factorize the following:

Question 1:  x^2+18x+81

Answer:

x^2+18x+81 = (x+9)^2

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Question 2:  a^2-14a+49 

Answer:

a^2-14a+49 = (a-7)^2

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Question 3:  4x^2+12x+9 

Answer:

4x^2+12x+9 = (2x+3)^2

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Question 4:  9x^2-24x+16 

Answer:

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

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Question 5:  49x^2+56x+16 

Answer:

49x^2+56x+16 = (7x+4)^2

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Question 6:  25z^2-30z+9   

Answer:

25z^2-30z+9 = (5z-3)^2

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Question 7:  9q^4 r^4-6p^4 q^2 r^2+p^8

Answer:

9q^4 r^4-6p^4 q^2 r^2+p^8 = (3q^2 r^2-p^4 )^2

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Question 8:  \displaystyle \frac{9p^2}{q^2} +\frac{16r^2}{m^2} +\frac{24pr}{qm}

Answer:

\displaystyle \frac{9p^2}{q^2} +\frac{16r^2}{m^2} +\frac{24pr}{qm} = (\frac{3p}{q}+\frac{4r}{m})^2

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Question 9:  \displaystyle \frac{1}{4} z^6+9a^2-3az^3 

Answer:

\displaystyle \frac{1}{4} z^6+9a^2-3az^3 = ( \frac{1}{2} z^3-3a)^2

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Question 10:  \displaystyle \frac{9}{4} a^2+ \frac{49}{9} p^2-7ap

Answer:

\displaystyle \frac{9}{4} a^2+ \frac{49}{9} p^2-7ap = ( \frac{3}{2} a- \frac{7}{3} p)^2

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Question 11:  t^2+22t+85 = 0

Answer:

Find two numbers with sum = 22 and product = 85

a+b = 22    &  ab = 85   \Rightarrow a =17,  b = 5

Hence the factors are

t^2+22t+85 = (t+17)(t+5)

Note: This technique would be used all across this exercise.

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Question 12:  x^2-10x+24

Answer:

a+b = -10   & ab = 24    \Rightarrow a = -6, b = -4

Therefore \ \ x^2-10x+24 = (x-6)(x-4)

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Question 13:  m^2-3m-40

Answer:

a+b = -3 & ab = -40   \Rightarrow a = -8, b = +5

Therefore \ \ m^2-3m-40 = (m-8)(m+5)

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Question 14:  x^2+x-72

Answer:

a+b = 1 & ab = -72   \Rightarrow a = 9, b = -8

Therefore \ \ x^2+x-72 = (x+9)(x-8)

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Question 15:  p^2-7p-120

Answer:

a+b = -7 & ab = -120   \Rightarrow a = 8, b = -15

Therefore \ \ p^2-7p-120 = (p+8)(p-15)

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Question 16:  16-17z+z^2

Answer:

a+b = -17 & ab = 16   \Rightarrow a = -1, b = -16

Therefore 16-17z+z^2 = (z-1)(z-16)

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Question 17:  a^2+5a-104

Answer:

x+y = 5 & xy = -104    \Rightarrow x = 13, y = -8

Therefore \ \ q^2+5a-104 = (a+13)(a-8)

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Question 18:  3x^2+11x+10

Answer:

a+b = 11 & ab = 30  \Rightarrow a = 5, b = 6

Therefore 3x^2+11x+10 = 3x^2+6x+5x+10

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

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

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Question 19:  6x^2+7x-3

Answer:

a+b = 7 & ab = -18   \Rightarrow a = 9, b = -2

Therefore 6x^2+7x-3 = 6x^2+9x-2x-3

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

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

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Question 20:  3z^2-4z-4

Answer:

a+b = -4 & ab = -12    \Rightarrow a = -6, b = 2

Therefore 3z^2-4z-4 = 3z^2-6z+2z-4

= 3z(z-2)+2(z-2)

= (z-2)(3z+2)

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Question 21:  72-x-x^2

Answer:

a+b = -1 & ab = -72   \Rightarrow a = -8, b = -9

Therefore 72-x-x^2 = -x^2-9x+8x+72

= -x(x-8)-9(x-8)

= (-x-9)(x-8)

= (x+9)(8-x)

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Question 22:  x^2-3xy-40y^2

Answer:

x^2-3xy-40y^2

= x^2-8xy+5xy-40y^2

= x(x+5y)-8y(x+5y)

= (x+5y)(x-8y)

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Question 23:  3x^2 y+11xy+6y

Answer:

a+b = 11 & ab = 18   \Rightarrow a = 9, b = 2

3x^2 y+11xy+6y

= y(3x^2+9x+2x+6)

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

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

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Question 24:  (a-b)^2-5(a-b)+6

Answer:

Let a-b = x

Hence (a-b)^2-5(a-b)+6 = x^2-5x+6 = (x-2)(x-3)

Substituting Back

= (a-b)^2-5(a-b)+6 = (a-b-2)(a-b-3)

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Question 25:  (a-3b)^2-4(a-3b)-21

Answer:

(a-3b)^2-4(a-3b)-21

= (a-3b)^2-7(a-3b)+3(a-3b)-21

= (a-3b)[(a-3b)-7]+3[(a-3b)-7]

= (a-3b-7)(a-3b)

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Question 26:  3(y-2)^2-(y-2)-44

Answer:

Let (y-2) = x

3(y-2)^2-(y-2)-44

= 3x^2-x-44 = 3x^2-12x+11x-44

= 3x(x-4)+11(x-4)

= (3x+11)(x-4)

Substituting Back

= (3(y-2)+11)(y-2-4)

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

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Question 27:  7+10(x+y)-8(x+y)^2

Answer:

Let (x+y) = a

Therefore 7+10(x+y)-8(x+y)^2

= 7+10a-8a^2

m+n = 10   &  mn = -56   \Rightarrow m = 14 \ \ n = -4

Therefore = 7+10a-8a^2+14a-4a+7

= -4a(2a+1)+7(2a+1)

= (7-4a)(2a+1)

Substituting Back

= (7-4x+4y)(8x+8y+1)

= 7+10(x+y)-8(x+y)^2 = (7-4x-4y)(8x+8y+1)