**Evaluate the following products:**

**(i) ax ^{2} ( bx + c) **

= ax^{2} (bx) + ax^{2} (c)

= abx^{3} + acx^{2}

**(ii) ab (a+b) **

= ab (a) + ab (b)

= a^{2}b + ab^{2}

**(iii) a**^{2}**b**^{2}** (ab**^{2}**+a**^{2}**b) **

= a^{2}b^{2} (ab^{2}) + a^{2}b^{2} (a^{2}b)

= a^{3}b^{4} + a^{4}b^{3}

**(iv) b ^{4}(b^{6} + b^{8}) **

= b^{4} (b^{6}) + b^{4} (b^{8})

= b^{4} (b^{6}) + b^{4} (b^{8})

= b^{10} + b^{12}

**Evaluate the following products:**

**(i) (x+3) (x+2) **

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

= x^{2} + 3x + 2x + 6

= x^{2} + 5x + 6

**(ii) (x+5) (x–2) **

= (x+5) x + (x+5) (–2)

= x^{2 }+5x – 2x –10

= x^{2} + 3x –10

**(iii) ( y – 4 ) ( y + 6 ) **

= (y – 4) y +(y – 4)6

= y^{2} – 4y + 6y – 24

= y^{2} + 2y – 24

**(iv) (a–5) (a–6) **

= (a–5) a + (a–6) (–6)

= a^{2} – 5a – 6a + 30

= a^{2} – 11a +30

**(v) (2x+1) (2x–3) **

= (2x+1)2x + (2x+1) (–3)

= 4x^{2} + 2x – 6x – 3x

= 4x^{2} – 4x –3

**(vi) ( a + b ) ( c + d ) **

= (a + b) c + (a+b) d

= ac + bc + ad + bd

**(vii) ( 2x – 3y ) ( x – y ) **

= (2x – 3y) x + (2x – 3y) (–y)

= 2x^{2} – 3xy – 2xy +3y^{2}

= 2x^{2} – 5xy + 3y^{2}

**(viii) ( √****𝟕𝐱**** + √****𝟓**** ) (√****𝟓𝐱**** + √****𝟕**** ) **

= (√𝟕𝐱 + √𝟓 ) (√𝟓𝐱) + (√𝟕𝐱 + √𝟓 ) (√𝟕)

= √𝟑𝟓 x^{2} + 5x + 7x + √𝟑𝟓

= √𝟑𝟓x^{2} + 12x + √𝟑𝟓

**(xi) (2a+3b) (2a–3b) **

= (2a+3b) 2a + (2a+3b) (–3b)

= 4a^{2}+ 6ab – 6ab – 9b^{2}

= 4a^{2} – 9b^{2}

**(xii) (6xy–5) (6xy+5) **

= (6xy – 5) (6xy) + (6xy – 5) 5

= 36x^{2}y^{2} – 30xy + 30xy – 25

= 36x^{2}y^{2} – 25

**(xiii)( ^{2}/_{x}+3) (^{2}/_{x} –7)**

= (^{2}/_{x}+3)^{2}/_{x} +(^{2}/_{x}+3)(-7)

= ^{4}/x^{2} + ^{6}/_{x} – ^{14}/_{x} – 21

= ^{4}/x^{2} – ^{8}/_{x} –21

**Expand the following using appropriate identity:**

**(i) (a +5) ^{2} **

Using (a + b)^{2} = a^{2} +2ab +b^{2} we get

a = a b = 5

(a +5)^{2} = a^{2} +2.a.5 +b^{2}

= a^{2} +10a +25

**(ii) (2a +3) ^{2} **

Using (a + b)^{2} = a^{2} +2ab +b^{2} we get

a = 2a; b = 3

(2a +3)^{2} = (2a)^{2} +2.2a.3 +b^{2}

= 4a^{2} + 12a + 9

**(iii) ( x + **^{𝟏}**/**_{𝐱}** ) ^{2} **

Using (a + b)^{2} = a^{2} +2ab +b^{2} we get

a = 2a; b = ^{1}/_{x}

(X + ^{𝟏}/ )^{2} = x^{2} + 2.x. 1x + (^{𝟏}/_{𝐱} )^{2}

= x^{2} + 2 + (^{𝟏}/x )^{2}

**(iv) ( √(12a) + √(6b) ) ^{2} **

Using (a + b)^{2} = a^{2} +2ab +b^{2} we get

a = √(12a) and b = √(6b)

(√(12a) + √ (6b))^{2}= (√12a)^{2} + 2.√12 a + √(6b) + (√(6b))^{2}

= 12a^{2} +2√72 ab +6b^{2}

= 12a^{2} +2 √ (36 ×2) ab +6b^{2}

= 12a^{2} + 12 √𝟐ab +6b^{2}

**(v) (****𝛑**** + **^{𝟐𝟐}**/ _{7} )2 **

Using (a + b)^{2} = a^{2} +2ab +b^{2} we get

a = 𝛑 and b = 22/7

(𝛑 + ^{22}/_{7} )^{2} = 𝛑^{2} + 2. 𝛑 ^{22}/_{7} + (^{22}/_{7})^{2}

= 𝛑^{2} + ^{44π}/_{7} + ( ^{22}/_{7})^{2 }

= 𝛑^{2} + ^{𝟒𝟒𝛑}/_{7} + ^{𝟒𝟖𝟒}/_{49}

**(vi) (y ****– 3) ^{2} **

Using (a – b)^{2} = a^{2} – 2ab + b^{2}

a = y and b = –3

(y – 3)^{2} = y^{2} – 2. y.3 + 3^{2}

= y^{2} – 6y + 9

**(vii) (3a – 2b) ^{2}**

Using (a – b)^{2} = a^{2} – 2ab + b^{2}

a = 3a and b = –2b

(3a – 2b)^{2} = (3a)^{2} – 2.3a.2b + (2b)^{2}

= 9a^{2} – 12ab + 4b^{2}

**(viii) ( y – **^{𝟏}**/ _{y} )^{2} **

Using (a – b)^{2} = a^{2} – 2ab + b^{2}

a = y and b = ^{𝟏}/_{y}

(y – ^{𝟏}/_{y})^{2} = y^{2} – 2.y. ^{𝟏}/_{y} + (^{𝟏}/_{y} )^{2}

= y^{2} – 2 + ^{𝟏}/_{y²}

**(ix) ( √(10x) – √(5y)) ^{2} **

Using (a – b)^{2} = a^{2} – 2ab + b^{2}

a = √(10x) and b = √(5y)

(√(10x) – √(5y))^{2}

= (√(10x))^{2} – 2. √(10x).√(5y)+ √(5y))^{2}

= 10x^{2} – 2√50 xy + 5y^{2}

= 10x^{2} – 2.5√2 xy + 5y^{2}

= 10x^{2} – 10√2 xy + 5y^{2}

**(x) (𝛑 – ^{𝟐𝟐}/_{7} )^{2} **

Using (a + b)^{2} = a^{2} +2ab +b^{2} we get

a = 𝛑 and b = ^{𝟐𝟐}/_{𝟕}

(𝛑 – ^{𝟐𝟐}/ )^{2} = 𝛑^{2} – 2. 𝛑 ^{𝟐𝟐}/_{𝟕} + (^{𝟐𝟐}/_{𝟕})^{2}

= 𝛑^{2} – ^{44}/_{𝟕} π + (^{𝟐𝟐}/_{𝟕})^{2}

= 𝛑^{2} – ^{𝟒𝟒𝛑}/_{7} + ^{𝟒𝟖𝟒}/_{𝟒𝟗}

**(xi) (2x+3) (2x+5) **

Using (x + a) (x + b) = x^{2} + x (a + b) ab we get

x = 2x, a = 3 and b = 5

(2x+3) (2x+5) = (2x)^{2} +2x (3 + 5) + 3.5

= 4x^{2} +16x +15

**(xii) (3x – 3) (3x + 4) **

Using (x + a) (x + b) = x^{2 }+ x (a + b) ab we get

x = 3x, a = –3 and b = 4

(3x – 3) (3x + 4) = (3x)^{2} + 3x [(–3)+(4)] + (-3)4

= 9x^{2} + 3x –12

= 9x^{2} + 3x –12

**Expand :**

**(i) (x + 3 ) (x – 3)**

Using (a + b) (a – b) = a^{2} – b^{2} we get

a = x, b = 3

(x + 3) (x – 3) = x^{2} – 3^{2}

= x^{2} – 9

** **

**(ii) (3x – 5y) (3x + 5y)**

Using (a + b) (a – b) = a^{2} – b^{2} we get

a = 3x, b = 5y

(3x – 5y) (3x + 5y) = (3x)^{2} – (5y)^{2}

= 9x^{2} – 25y^{2}

**(iii) ( ^{x}/_{3}+^{y}/_{2})( ^{x}/_{3}– ^{y}/_{2}) **

Using (a + b) (a – b) = a^{2} – b^{2} we get

a = ^{x}/_{3}, b = ^{y}/_{2}

(^{x}/_{3}+^{ y}/_{2})( ^{x}/_{3}– ^{y}/_{2})= (^{x}/_{3} )^{2} – (^{x}/_{3})^{2}

= 𝐱^{𝟐}/_{9-}𝐲^{2}/_{𝟒}

**(iv) (x ^{2} + y^{2}) (x^{2} – y^{2}) **

Using (a + b) (a – b) = a^{2} – b^{2} we get

a = x^{2}, b = y^{2}

(x^{2} + y^{2}) (x^{2} – y^{2}) = (x^{2})^{2} – (y^{2})^{2}

= x^{4} – y^{4}

**(v) (a ^{2} + 4b^{2}) (a + 2b) (a – 2b) **

Using (a + b) (a – b) = a^{2} – b^{2} for 2nd and 3rd term we get

(a^{2} + 4b^{2}) (a + 2b) (a – 2b) = (a^{2} + 4b^{2}) [a^{2} – (2b)^{2}]

= (a^{2} + 4b^{2}) (a^{2} – 4b^{2})

Using the above identity once again we get

= (a^{2})^{2} – (4b^{2})^{2}

= a^{4} – 16b^{4}

**(vi) (x – 4) (x + 4) (x – 3) (x + 4) **

Using (a + b) (a – b) = a^{2} – b^{2} we get

(x – 4) (x + 4) (x – 3) (x + 4) = (x^{2} – 4^{2}) (x^{2} – 3^{2})

= (x^{2} – 16) (x^{2} – 9)

Using (a + b) (a – b) = x^{2} – x (a + b) + ab

= (x^{2})^{2} – x (16+9) +16×9

= x^{4} – 25x^{2} +144

**(vii) (x – a) (x + a)(**^{𝟏}**/ _{𝐱}**

**−**

^{𝟏}**/**

_{𝐚}**)(**

^{𝟏}**/**

_{𝐱}**+**

^{𝟏}**/**

_{𝐚}**)**

(x^{2} – a^{2}) )(( /_{𝐱} )^{2}– (^{1}/_{a })^{2})

x^{2} x ^{𝟏}/_{𝐱}^{ 2} – x^{2} x ^{𝟏}/_{𝐚} ^{2 }– a^{2} x ^{𝟏}/_{𝐱}^{ 2} + a^{2} x ^{𝟏}/_{𝐚} ^{2}

1 – x^{2}/a^{2} − a^{2}/ x^{2} + 1

2 – 𝐱^{𝟐}/a^{𝟐} − 𝐚^{𝟐} /x^{𝟐}

**Simplify the following:**

**(i) (2x – 3y) ^{2 }+ 12xy **

= (2x)^{2} + (3y)^{2} – 2.2x.3y + 12xy

= 4x^{2} + 9y^{2} -12xy +12xy

= 4x^{2} + 9y^{2}

**(ii) (3m + 5n) ^{2} – (2n)^{2} **

= (3m)^{2} + (5n)^{2} + 2.3m.5n – 4n^{2}

= 9m^{2} + 25n^{2} + 30mn – 4n^{2}

= 9m^{2} + 30 mn +21n^{2}

**(iii) (4a – 7b) ^{2} – (3a)^{2} **

= (4a)^{2} – 2.4a.7b + (7b)^{2} – (3a)^{2}

= 16a^{2} – 56ab + 49b^{2} – 9a^{2}

= 7a^{2} -56ab + 49b^{2}

**(iv) (x + **^{𝟏}**/ _{x})^{2} (m + ^{𝟏}**

**/**

_{𝐦}**)**

^{2}= (x^{2} + 2. x. ^{𝟏}/_{x} + ^{𝟏}/_{x}^{2})^{2} – (m^{2} + 2. m. ^{𝟏}/_{𝐦} + ^{𝟏}/_{𝐦} ^{2})^{2}

= x^{2} + 2 + ^{𝟏}/_{x}^{ 2} – (m^{2} + 2 + ^{𝟏}/_{𝐦} ^{2})

= x^{2} + 2 + ^{𝟏}/_{x}^{ 2} – m^{2} + ^{2} – ^{𝟏}/_{𝐦} ^{2}

= x^{2} – m^{2}+ ^{𝟏}/_{x} ^{𝟐} – ^{𝟏}/_{𝐦} ^{𝟐} + 4

**(v) (m ^{2} + 2n^{2})^{2} – 4m^{2}n^{2} **

= m^{4 }+2m^{4}.2n^{2} +4n^{4} – 4m^{2} n^{2}

= m^{4}+ 4m^{2}n^{2 }+ 4n^{2 }– 4m^{2}n^{2 }

= m^{4} + 4n^{2}

**(vi) (3a – 2) ^{2} – (2a -3)^{2} **

= (9a^{2} – 2.3a.2 + 2^{2}) – (4a^{2} – 2.3a.2 + 9^{2})

= 9a^{2} –12a + 4 – 4a^{2} + 12a – 9

= 5a^{2} – 5

=5(a^{2} – 1)

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