Algebra - The Difference Of Two Squares

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Question 1


Expand, notice any patterns:

a) $(x+4)(x-4)$a s v d
b) $(x+8)(x-8)$a s v d
c) $(x-3)(x+3)$a s v d
d) $(z-10)(z+10)$a s v d
e) $(9+x)(9-x)$a s v d
f) $(2x+3)(2x-3)$a s v d
g) $(5-3y)(5+3y)$a s v d
h) $2(x+6)(x-6)$a s v d
i) $(4z+3x)(4z-3x)$a s v d
j) $(x^n+2)(x^n-2)$a s v d
k) $4x(5-6x)(5+6x)$a s v d
l) $(x+y+11)(x+y-11)$a s v d


Question 2


Factorise the following using the difference of two squares:

a) $x^2-25$a s v d
b) $x^2-49$a s v d
c) $x^2-81$a s v d
d) $x^2-144$a s v d
e) $x^2-1$a s v d
f) $z^2-4$a s v d
g) $y^2-100$a s v d
h) $a^2-16$a s v d


Question 3


By taking out a common factor first, factorise the following:

a) $2x^2-18$a s v d
b) $3x^2-75$a s v d
c) $5x^2-500$a s v d
d) $x^3-64x$a s v d
e) $x^4-36x^2$a s v d
f) $2x^3-8x$a s v d
g) $7x^3-7x$a s v d
h) $9x^4-144x^2$a s v d


Question 4


Factorise:

a) $4x^2-9$a s v d
b) $9x^2-4$a s v d
c) $25x^2-16$a s v d
d) $16x^2-1$a s v d
e) $49x^2-25$a s v d
f) $x^4-9$a s v d
g) $x^6-1$a s v d
h) $x^{10}-49$a s v d
i) $9x^4-25$a s v d
j) $25x^8-81$a s v d


Question 5


Factorise the following, you may need to take out a common factor first:

a) $9x^3-16x$a s v d
b) $16z^3-z$a s v d
c) $9-4x^2$a s v d
d) $64-25x^2$a s v d
e) $18x^2-2$a s v d
f) $125x^2-20$a s v d
g) $36x^3-100x$a s v d
h) $12x-27x^3$a s v d


Question 6


Factorise:

a) $y^2-z^2$a s v d
b) $a^2-b^2$a s v d
c) $4x^2-y^2$a s v d
d) $9y^2-4x^2$a s v d
e) $2x^2-18z^2$a s v d
f) $x^2z-y^2z$a s v d
g) $48x^2-75y^2$a s v d
h) $45x^3-80xy^2$a s v d


Question 7


Factorise:

a) $x^2-\frac19$a s v d
b) $y^2-\frac49$a s v d
c) $9x^2-\frac14$a s v d
d) $36x^2-\frac{25}{49}$a s v d
e) $\frac14x^2-\frac{1}{16}$a s v d
f) $\frac {x^2}{9}-\frac{4}{25}$a s v d
g) $\frac {9x^2}{4}-\frac{36y^2}{100}$a s v d
h) $\frac {125}{9}a^2-\frac{405}{64}b^2$a s v d


Question 8


Expand, leaving answers in the form $ax^2+bx+c$:

a) $((x+2)+3)((x+2)-3)$a s v d
b) $((x+1)+5)((x+1)-5)$a s v d
c) $((x+3)+8)((x+2)-8)$a s v d
d) $(x+y-5)(x+y+5)$a s v d
e) $(5+2x+y)(5-(2x+y))$a s v d
f) $(9+x+y)(9-x-y)$a s v d


Question 9


Factorise completely:

a) $x^4-1$a s v d
b) $16x^4-1$a s v d
c) $x^8-256$a s v d
d) $243x^4-48$a s v d


Question 10


Use the difference of two squares and irrational numbers to factorise the following:

a) $x^2-2$a s v d
b) $x^2-5$a s v d
c) $x^2-8$a s v d
d) $y^2-20$a s v d
e) $a^6-150$a s v d
f) $9p^{10}-98q^2$a s v d


Question 11


Factorise and simplify fully:

a) $(x+y)^2-9$a s v d
b) $(x-3)^2-25$a s v d
c) $(2x-1)^2-100$a s v d
d) $(5x-2)^2-9x^2$a s v d
e) $16x^2-(10x-3)^2$a s v d
f) $(4x-1)^2-(3x-5)^2$a s v d
g) $25(x+5)^2-100(2x-3)^2$a s v d
h) $49(3x-2)^4-4(7x-1)^2$a s v d