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Let’s factorise.
a) $4a^2 + 4a + 1$
b) $4a^2 + 20a + 25$
c) $9a^2 + 6a + 1$
d) $9x^2 + 12x + 4$
e) $9a^2 + 24a + 16$
f) $16x^2 + 24x + 9$
g) $9x^2 – 24x + 16$
h) $25x^2 – 80x + 64$
i) $16a^2 – 40a + 25$
j) $49p^2 – 14p + 1$
k) $36 – 60p + 25p^2$
l) $144 – 120x + 25x^2$
Let’s resolve into factors.
a) $x^2 + 6x + 9$
b) $x^2 + 8x + 16$
c) $x^2 + 10x + 25$
d) $a^2 + 12a + 36$
e) $a^2 + 14a + 49$
f) $a^2 + 16a + 64$
g) $x^2 – 4x + 4$
h) $x^2 – 2x + 1$
i) $p^2 – 18p + 81$
j) $p^2 – 20p + 100$
k) $p^2 – 24p + 144$
l) $x^2 – 30x + 225$
Let’s find the expressions to be inserted to make the following expressions perfect squares.
a) $x^2 +$ ……. $+ 4$
b) $x^2 +$ ……. $+ 9$
c) $a^2 -$ ……. $+ 16$
d) $a^2 -$ ……. $+ 64$
e) $9x^2 +$ ……. $+ 4y^2$
f) $25x^2 -$ ……. $+ 81y^2$
Let’s rewrite these expressions in the forms of $a^2 + 2ab + b^2$ or $a^2 – 2ab + b^2$.
E.g.
$x^2 + 2x + 1 = x^2 + 2.x.1 + 1^2$
$9x^2 – 30x + 25 = (3x)^2 – 2.3x.5 + 5^2$
a) $x^2 + 4x + 4 =$ ………………..
b) $x^2 – 8x + 16 =$ ………………..
c) $4x^2 + 12x + 9 =$ ………………..
d) $9x^2 – 12x + 4 =$ ………………..
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Let’s express the following algebraic expressions as perfect squares.
a) $x^2 + 10x + 25$
b) $a^2 + 20a + 100$
c) $4x^2 + 28x + 49$
d) $4x^2 + 12xy + 9y^2$
e) $a^2 – 6a + 9$
f) $x^2 – 16x + 64$
g) $9m^2 – 12mn + 4n^2$
h) $25a^2 – 20ab + 4b^2$
Simplify.
a) $frac{2.1 times 2.1 – 0.9 times 0.9}{2.1 – 0.9}$
b) $frac{3.6 times 3.6 – 1.4 times 1.4}{3.6 + 1.4}$
c) $frac{2.5 times 2.5 – 1.4 times 1.4}{2.5 + 1.4}$
d) $frac{5.6 times 5.6 – 4.4 times 4.4}{5.6 – 4.4}$
Let’s say and write the following algebraic expressions as the perfect square forms.
E.g.
$a^2 + 2.a.b + b^2 = (a + b)^2$
$(3x)^2 – 2.3x.5 + 5^2 = (3x – 5)^2$
| Algebraic Expressions | Perfect Square Forms |
| a) $x^2 + 2.x.y + y^2$ | |
| b) $m^2 + 2.m.1 + 1^2$ | |
| c) $p^2 + 2.p.3 + 3^2$ | |
| d) $(2a)^2 + 2.2a.3b + (3b)^2$ | |
| e) $m^2 – 2.m.n + n^2$ | |
| f) $x^2 – 2.x.5 + 5^2$ | |
| g) $(2x)^2 – 2.2x.7 + 7^2$ | |
| h) $(3x)^2 – 2.3x.5y + (5y)^2$ |
Let’s simplify by factorisation process.
a) $25^2 – 15^2$
b) $64^2 – 44^2$
c) $96^2 – 86^2$
d) $(101)^2 – (100)^2$
e) $51 times 49$
f) $82 times 78$
g) $101 times 99$
h) $103 times 97$
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Let’s factorise:
a) $(a – b)^2 – 9$
b) $(x + y)^2 – 25$
c) $(a^2 + b^2)^2 – 4$
d) $(a^2 – b^2)^2 – a^2b^2$
e) $16 – (x + y)^2$
f) $49 – (a – b)^2$
g) $(x + 3)^2 – (y + 2)^2$
h) $(a – 5)^2 – (b – 4)^2$
Let’s factorise:
a) $x^4 – 16$
b) $a^4 – 81$
c) $x^4 – y^4$
d) $16p^4 – q^4$
e) $81x^4 – 625$
f) $32y^4 – 162$
g) $a^8 – b^8$
h) $256x^8 – y^8$
