Factorize $$4x^2 - 12x + 9$$

$$\left( 2x^2 - 6x + \dfrac{9}{2}\right)^2$$

Which alternative below is equal to $$50 \times 52$$?

$$51^2 - 2\times 51 \times 1 + 1^2$$

$$51^2 + 2\times 51 \times 1 + 1^2$$

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>Parentheses>Activities 2

At the bottom of the page there is an interactive quiz.

Activities 2

2.1)

Expand the parentheses:

a): ${(x + 1)}^{2}$
b): ${(x + 2)}^{2}$
c): ${(x + 3)}^{2}$
d): ${(x + 4)}^{2}$
e): ${(x + 5)}^{2}$
f): ${(x + 6)}^{2}$
g): ${(x + 7)}^{2}$
h): ${(x + 8)}^{2}$
i): ${(x + 9)}^{2}$

a): ${(x + 1)}^{2}$
b): ${(x + 2)}^{2}$
c): ${(x + 3)}^{2}$
d): ${(x + 4)}^{2}$
e): ${(x + 5)}^{2}$
f): ${(x + 6)}^{2}$
g): ${(x + 7)}^{2}$
h): ${(x + 8)}^{2}$
i): ${(x + 9)}^{2}$

2.2)

Expand the parentheses:

a): $x^{2} - 12 x + 36$
b): $x^{2} + 6 x + 9$
c): $x^{2} - 18 x + 81$
d): $x^{2} - 2 x + 1$
e): $x^{2} + 4 x + 4$
f): $x^{2} - 14 x + 49$
g): $x^{2} + 16 x + 64$
h): $x^{2} - 10 x + 25$
i): $x^{2} - 8 x + 16$

a): $x^{2} - 12 x + 36$
b): $x^{2} + 6 x + 9$
c): $x^{2} - 18 x + 81$
d): $x^{2} - 2 x + 1$
e): $x^{2} + 4 x + 4$
f): $x^{2} - 14 x + 49$
g): $x^{2} + 16 x + 64$
h): $x^{2} - 10 x + 25$
i): $x^{2} - 8 x + 16$

2.3)

Factorize the expressions:

a): $x^{2} - 12 x + 36$
b): $x^{2} + 6 x + 9$
c): $x^{2} - 18 x + 81$
d): $x^{2} - 2 x + 1$
e): $x^{2} + 4 x + 4$
f): $x^{2} - 14 x + 49$
g): $x^{2} + 16 x + 64$
h): $x^{2} - 10 x + 25$
i): $x^{2} - 8 x + 16$

a): $x^{2} - 12 x + 36$
b): $x^{2} + 6 x + 9$
c): $x^{2} - 18 x + 81$
d): $x^{2} - 2 x + 1$
e): $x^{2} + 4 x + 4$
f): $x^{2} - 14 x + 49$
g): $x^{2} + 16 x + 64$
h): $x^{2} - 10 x + 25$
i): $x^{2} - 8 x + 16$

2.4)

Expand the parentheses:

a): ${(x + 2)}^{2}$
b): ${(x + 3)}^{2}$
c): ${(3 x + 4)}^{2}$
d): ${(2 x + 3)}^{2}$

a): ${(x + 2)}^{2}$
b): ${(x + 3)}^{2}$
c): ${(3 x + 4)}^{2}$
d): ${(2 x + 3)}^{2}$

2.5)

Expand the parentheses:

a): ${(x + 9)}^{2}$
b): ${(1 + x)}^{2}$
c): ${(8 + 5 x)}^{2}$
d): ${(x + \frac{1}{2})}^{2}$
e): ${(3 a + 4 b)}^{2}$
f): ${(2 a + 5 b)}^{2}$

a): ${(x + 9)}^{2}$
b): ${(1 + x)}^{2}$
c): ${(8 + 5 x)}^{2}$
d): ${(x + \frac{1}{2})}^{2}$
e): ${(3 a + 4 b)}^{2}$
f): ${(2 a + 5 b)}^{2}$

2.6)

Expand the parentheses:

a): ${(x - 2)}^{2}$
b): ${(x - 3)}^{2}$
c): ${(5 x - 3)}^{2}$
d): ${(2 x - 4)}^{2}$

a): ${(x - 2)}^{2}$
b): ${(x - 3)}^{2}$
c): ${(5 x - 3)}^{2}$
d): ${(2 x - 4)}^{2}$

2.7)

Expand the parentheses:

a): ${(x - 13)}^{2}$
b): ${(1 - x)}^{2}$
c): ${(3 a - 4 b)}^{2}$
d): ${(x - \frac{3}{4})}^{2}$
e): ${(\frac{x}{3} - \frac{2}{3})}^{2}$
f): ${(2 x - \frac{5}{4})}^{2}$

a): ${(x - 13)}^{2}$
b): ${(1 - x)}^{2}$
c): ${(3 a - 4 b)}^{2}$
d): ${(x - \frac{3}{4})}^{2}$
e): ${(\frac{x}{3} - \frac{2}{3})}^{2}$
f): ${(2 x - \frac{5}{4})}^{2}$

2.8)

Factorize the expressions:

a): $x^{2} - 1$
b): $4 x^{2} - 25$
c): $2 x^{2} - 18$
d): $3 x^{2} - 48 x$
e): $18 - 2 x^{2}$
f): $7 x^{2} - 21$

a): $x^{2} - 1$
b): $4 x^{2} - 25$
c): $2 x^{2} - 18$
d): $3 x^{2} - 48 x$
e): $18 - 2 x^{2}$
f): $7 x^{2} - 21$

2.9)

Expand the parentheses:

a): $(x - 2) (x + 2)$
b): $(x + 3) (x - 3)$
c): $(4 x + 3) (4 x - 3)$
d): $(2 x + 4) (2 x - 4)$

a): $(x - 2) (x + 2)$
b): $(x + 3) (x - 3)$
c): $(4 x + 3) (4 x - 3)$
d): $(2 x + 4) (2 x - 4)$

2.10)

Expand the parentheses:

a): $(x - 7) (x + 7)$
b): $(x - 11) (x + 11)$
c): $(3 - x) (3 + x)$
d): $(2 a + b) (2 a - b)$
e): $(3 a + 4 b) (3 a - 4 b)$
f): $(\frac{1}{2} + \frac{1}{3} a) (\frac{1}{2} - \frac{1}{3} a)$

a): $(x - 7) (x + 7)$
b): $(x - 11) (x + 11)$
c): $(3 - x) (3 + x)$
d): $(2 a + b) (2 a - b)$
e): $(3 a + 4 b) (3 a - 4 b)$
f): $(\frac{1}{2} + \frac{1}{3} a) (\frac{1}{2} - \frac{1}{3} a)$

2.11)

Factorize the fractions and simplify if possible:

a): $\frac{x^{2} - 25}{x + 5}$
b): $\frac{x^{2} - 81}{3 x + 27}$
c): $\frac{100 x^{2} - 1}{10 x - 1}$
d): $\frac{2 a^{2} - 50}{18 a - 90}$
e): $\frac{2 x^{2} - 8}{x - 2}$
f): $\frac{x^{2} - \frac{1}{4}}{2 x - 1}$
g): $\frac{x^{2} - 4 x + 4}{x - 2}$

a): $\frac{x^{2} - 25}{x + 5}$
b): $\frac{x^{2} - 81}{3 x + 27}$
c): $\frac{100 x^{2} - 1}{10 x - 1}$
d): $\frac{2 a^{2} - 50}{18 a - 90}$
e): $\frac{2 x^{2} - 8}{x - 2}$
f): $\frac{x^{2} - \frac{1}{4}}{2 x - 1}$
g): $\frac{x^{2} - 4 x + 4}{x - 2}$

2.12)

Expand the parentheses and contract the resulting expressions:

a): ${(x + 3)}^{2} + {(x - 2)}^{2}$
b): ${(x + 2)}^{2} + {(4 - x)}^{2}$
c): ${(x + 5)}^{2} + {(x - 1)}^{2}$
d): ${(x + 4)}^{2} + {(x + 8)}^{2}$
e): ${(x + 7)}^{2} + {(x + 7)}^{2}$
f): ${(x + 6)}^{2} - {(x + 6)}^{2}$
g): ${(x + 9)}^{2} - {(x - 3)}^{2}$
h): ${(x + 8)}^{2} + {(5 - x)}^{2}$
i): ${(x + 9)}^{2} + {(x + 10)}^{2}$

a): ${(x + 3)}^{2} + {(x - 2)}^{2}$
b): ${(x + 2)}^{2} + {(4 - x)}^{2}$
c): ${(x + 5)}^{2} + {(x - 1)}^{2}$
d): ${(x + 4)}^{2} + {(x + 8)}^{2}$
e): ${(x + 7)}^{2} + {(x + 7)}^{2}$
f): ${(x + 6)}^{2} - {(x + 6)}^{2}$
g): ${(x + 9)}^{2} - {(x - 3)}^{2}$
h): ${(x + 8)}^{2} + {(5 - x)}^{2}$
i): ${(x + 9)}^{2} + {(x + 10)}^{2}$

2.13)

Expand the parentheses and contract the resulting expressions:

a): ${(x + 3)}^{2}$
b): ${(a - 5)}^{2}$
c): $(x + 3) (x - 3)$
d): $2 {(x + 4)}^{2}$
e): ${(x - \frac{1}{3})}^{2}$
f): $\begin{aligned} {(x + 2)}^{2} + {(x - 3)}^{2} \\ + (x + 2) (x - 2) \end{aligned}$
g): $\begin{aligned} (4 - a) (4 + a) \\ - {(a + 2)}^{2} + {(a - 2)}^{2} \end{aligned}$
h): $- 3 {(2 - x)}^{2} - 5 {(3 x - 4)}^{2} + 2 x^{2}$
i): $\begin{aligned} - (4 - x) (4 + x) - {(2 x - 5)}^{2} \\ - {(x - \frac{1}{3})}^{2} \end{aligned}$
j): $\begin{aligned} (2 - a) (2 + a) - 2 {(a + 1)}^{2} \\ - 9 {(a - \frac{1}{3})}^{2} \end{aligned}$
k): $\begin{aligned} 2 (2 a - 1) (2 a + 1) - {(a - 2)}^{2} \\ - 4 {(a + \frac{1}{2})}^{2} \end{aligned}$
l): $\begin{aligned} {(a - \frac{1}{2})}^{2} + \frac{1}{4} {(2 a + 1)}^{2} \\ - \frac{1}{2} (2 a - 1) (2 a + 1) \end{aligned}$

a): ${(x + 3)}^{2}$
b): ${(a - 5)}^{2}$
c): $(x + 3) (x - 3)$
d): $2 {(x + 4)}^{2}$
e): ${(x - \frac{1}{3})}^{2}$
f): ${(x + 2)}^{2} + {(x - 3)}^{2} + (x + 2) (x - 2)$
g): $(4 - a) (4 + a) - {(a + 2)}^{2} + {(a - 2)}^{2}$
h): $- 3 {(2 - x)}^{2} - 5 {(3 x - 4)}^{2} + 2 x^{2}$
i): $- (4 - x) (4 + x) - {(2 x - 5)}^{2} - {(x - \frac{1}{3})}^{2}$
j): $(2 - a) (2 + a) - 2 {(a + 1)}^{2} - 9 {(a - \frac{1}{3})}^{2}$
k): $2 (2 a - 1) (2 a + 1) - {(a - 2)}^{2} - 4 {(a + \frac{1}{2})}^{2}$
l): ${(a - \frac{1}{2})}^{2} + \frac{1}{4} {(2 a + 1)}^{2} - \frac{1}{2} (2 a - 1) (2 a + 1)$

2.14)

Expand the parentheses and contract the resulting expressions:

a): ${(1 + \frac{a}{2})}^{2}$
b): ${(x + \frac{1}{x})}^{2}$
c): $(\sqrt{x} + 1) (\sqrt{x} - 1)$
d): ${(x - 3)}^{2} + (x - 2) (x + 2)$

a): ${(1 + \frac{a}{2})}^{2}$
b): ${(x + \frac{1}{x})}^{2}$
c): $(\sqrt{x} + 1) (\sqrt{x} - 1)$
d): ${(x - 3)}^{2} + (x - 2) (x + 2)$

2.15)

Factorize the expressions:

a): $x^{2} - 81$
b): $x^{2} - 1$
c): $a^{2} - 100$
d): $25 - x^{2}$
e): $4 x^{2} - 16$
f): $49 - 9 a^{2}$

a): $x^{2} - 81$
b): $x^{2} - 1$
c): $a^{2} - 100$
d): $25 - x^{2}$
e): $4 x^{2} - 16$
f): $49 - 9 a^{2}$

2.16)

Factorize the expressions:

a): $x^{2} - 4 x + 4$
b): $x^{2} - 36$
c): $2 x^{3} - 4 x^{2} + 2 x$
d): $2 y^{2} + 20 y + 50$
e): $3 x^{4} - 12 x^{2}$
f): $5 a^{2} - 20$

a): $x^{2} - 4 x + 4$
b): $x^{2} - 36$
c): $2 x^{3} - 4 x^{2} + 2 x$
d): $2 y^{2} + 20 y + 50$
e): $3 x^{4} - 12 x^{2}$
f): $5 a^{2} - 20$

2.17)

Factorize the fractions and simplify if possible:

a): $\frac{a^{2} - 49}{a + 7}$
b): $\frac{x + 8}{x^{2} - 64}$
c): $\frac{25 a^{2} - 81}{5 a - 9}$
d): $\frac{16 a^{2} - 36}{6}$
e): $\frac{9 x^{2} - 4}{3 x - 2}$
f): $\frac{9 a^{2} - 64}{6 a + 16}$

2.18)

Factorize the expressions:

a): $2 a^{2} - 50$
b): $12 x^{2} - 27$
c): $20 a^{2} - 5$
d): ${(x + 5)}^{2} - 81$
e): $- 4 + 25 x^{2}$
f): $20 a^{2} - 80$
g): ${(2 a - 4)}^{2} - 9 a^{2}$
h): ${(x - 3)}^{2} - {(2 x + 2)}^{2}$

a): $2 a^{2} - 50$
b): $12 x^{2} - 27$
c): $20 a^{2} - 5$
d): ${(x + 5)}^{2} - 81$
e): $- 4 + 25 x^{2}$
f): $20 a^{2} - 80$
g): ${(2 a - 4)}^{2} - 9 a^{2}$
h): ${(x - 3)}^{2} - {(2 x + 2)}^{2}$

2.19)

Simplify the expressions by using the algebraic identities:

a): $\frac{{(x + 2)}^{2}}{x^{2} - 4}$
b): $\frac{2 x^{2} - 8}{x - 2}$
c): $\frac{x^{2} - 9}{2 x + 6}$
d): $\frac{x}{x^{2} - 4} - \frac{1}{x + 2}$
e): $\frac{x + 1}{x - 3} - \frac{x^{2} + 15}{x^{2} - 9}$

a): $\frac{{(x + 2)}^{2}}{x^{2} - 4}$
b): $\frac{2 x^{2} - 8}{x - 2}$
c): $\frac{x^{2} - 9}{2 x + 6}$
d): $\frac{x}{x^{2} - 4} - \frac{1}{x + 2}$
e): $\frac{x + 1}{x - 3} - \frac{x^{2} + 15}{x^{2} - 9}$

2.20)

Factorize the fractions and simplify them if possible:

a): $\frac{x - 1}{x^{2} - 1}$
b): $\frac{4 x^{2} - 8 x + 4}{2 x - 2}$
c): $\frac{x^{2} + 4 x + 4}{4 - x^{2}}$
d): $\frac{5 - x^{2}}{\sqrt{5} - x}$
e): $\frac{{(x + 2 y)}^{2}}{x^{2} - 4 y^{2}}$

2.21)

Solve without using a calculator:

Example 1

\begin{array}{l} 99 \times 101 & = (100 - 1) (100 + 1) \\ = 10 0^{2} - 1^{2} \\ = 10 000 - 1 \\ = 9999 \end{array}

Find the answers by using the third algebraic identity as illustrated in Example 1:

a): $29 \times 31$
b): $18 \times 22$
c): $25 \times 15$
d): $103 \times 97$

a): $29 \times 31$
b): $18 \times 22$
c): $25 \times 15$
d): $103 \times 97$

You can find the solutions here.

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Applications of the Algebraic Identities

Solutions