# Activities 2

## 2.1)

Expand the parentheses:

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

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

## 2.2)

Expand the parentheses:

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

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

## 2.3)

Factorize the expressions:

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

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

## 2.4)

Expand the parentheses:

a)
${\left(x+2\right)}^{2}$
b)
${\left(x+3\right)}^{2}$
c)
${\left(3x+4\right)}^{2}$
d)
${\left(2x+3\right)}^{2}$

a)
${\left(x+2\right)}^{2}$
b)
${\left(x+3\right)}^{2}$
c)
${\left(3x+4\right)}^{2}$
d)
${\left(2x+3\right)}^{2}$

## 2.5)

Expand the parentheses:

a)
${\left(x+9\right)}^{2}$
b)
${\left(1+x\right)}^{2}$
c)
${\left(8+5x\right)}^{2}$
d)
$\phantom{\rule{-0.17em}{0ex}}{\left(x+\frac{1}{2}\right)}^{2}$
e)
${\left(3a+4b\right)}^{2}$
f)
${\left(2a+5b\right)}^{2}$

a)
${\left(x+9\right)}^{2}$
b)
${\left(1+x\right)}^{2}$
c)
${\left(8+5x\right)}^{2}$
d)
$\phantom{\rule{-0.17em}{0ex}}{\left(x+\frac{1}{2}\right)}^{2}$
e)
${\left(3a+4b\right)}^{2}$
f)
${\left(2a+5b\right)}^{2}$

## 2.6)

Expand the parentheses:

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

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

## 2.7)

Expand the parentheses:

a)
${\left(x-13\right)}^{2}$
b)
${\left(1-x\right)}^{2}$
c)
${\left(3a-4b\right)}^{2}$
d)
$\phantom{\rule{-0.17em}{0ex}}{\left(x-\frac{3}{4}\right)}^{2}$
e)
$\phantom{\rule{-0.17em}{0ex}}{\left(\frac{x}{3}-\frac{2}{3}\right)}^{2}$
f)
$\phantom{\rule{-0.17em}{0ex}}{\left(2x-\frac{5}{4}\right)}^{2}$

a)
${\left(x-13\right)}^{2}$
b)
${\left(1-x\right)}^{2}$
c)
${\left(3a-4b\right)}^{2}$
d)
$\phantom{\rule{-0.17em}{0ex}}{\left(x-\frac{3}{4}\right)}^{2}$
e)
$\phantom{\rule{-0.17em}{0ex}}{\left(\frac{x}{3}-\frac{2}{3}\right)}^{2}$
f)
$\phantom{\rule{-0.17em}{0ex}}{\left(2x-\frac{5}{4}\right)}^{2}$

## 2.8)

Factorize the expressions:

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

## 2.9)

Expand the parentheses:

a)
$\left(x-2\right)\left(x+2\right)$
b)
$\left(x+3\right)\left(x-3\right)$
c)
$\left(4x+3\right)\left(4x-3\right)$
d)
$\left(2x+4\right)\left(2x-4\right)$

a)
$\left(x-2\right)\left(x+2\right)$
b)
$\left(x+3\right)\left(x-3\right)$
c)
$\left(4x+3\right)\left(4x-3\right)$
d)
$\left(2x+4\right)\left(2x-4\right)$

## 2.10)

Expand the parentheses:

a)
$\left(x-7\right)\left(x+7\right)$
b)
$\left(x-11\right)\left(x+11\right)$
c)
$\left(3-x\right)\left(3+x\right)$
d)
$\left(2a+b\right)\left(2a-b\right)$
e)
$\left(3a+4b\right)\left(3a-4b\right)$
f)
$\phantom{\rule{-0.17em}{0ex}}\left(\frac{1}{2}+\frac{1}{3}a\right)\phantom{\rule{-0.17em}{0ex}}\left(\frac{1}{2}-\frac{1}{3}a\right)$

a)
$\left(x-7\right)\left(x+7\right)$
b)
$\left(x-11\right)\left(x+11\right)$
c)
$\left(3-x\right)\left(3+x\right)$
d)
$\left(2a+b\right)\left(2a-b\right)$
e)
$\left(3a+4b\right)\left(3a-4b\right)$
f)
$\phantom{\rule{-0.17em}{0ex}}\left(\frac{1}{2}+\frac{1}{3}a\right)\phantom{\rule{-0.17em}{0ex}}\left(\frac{1}{2}-\frac{1}{3}a\right)$

## 2.11)

Factorize the fractions and simplify if possible:

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

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

## 2.12)

Expand the parentheses and contract the resulting expressions:

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

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

## 2.13)

Expand the parentheses and contract the resulting expressions:

a)
${\left(x+3\right)}^{2}$
b)
${\left(a-5\right)}^{2}$
c)
$\left(x+3\right)\left(x-3\right)$
d)
$2{\left(x+4\right)}^{2}$
e)
$\phantom{\rule{-0.17em}{0ex}}{\left(x-\frac{1}{3}\right)}^{2}$
f)
$\begin{array}{ccc}\hfill & {\left(x+2\right)}^{2}+{\left(x-3\right)}^{2}\hfill & \hfill \\ \hfill & \phantom{\rule{2em}{0ex}}+\left(x+2\right)\left(x-2\right)\hfill \end{array}$
g)
$\begin{array}{ccc}\hfill & \left(4-a\right)\left(4+a\right)\hfill & \hfill \\ \hfill & \phantom{\rule{2em}{0ex}}-{\left(a+2\right)}^{2}+{\left(a-2\right)}^{2}\hfill \end{array}$
h)
$-3{\left(2-x\right)}^{2}-5{\left(3x-4\right)}^{2}+2{x}^{2}$
i)
$\begin{array}{ccc}\hfill & -\left(4-x\right)\left(4+x\right)-{\left(2x-5\right)}^{2}\hfill & \hfill \\ \hfill & \phantom{\rule{2em}{0ex}}-\phantom{\rule{-0.17em}{0ex}}{\left(x-\frac{1}{3}\right)}^{2}\hfill \end{array}$
j)
$\begin{array}{ccc}\hfill & \left(2-a\right)\left(2+a\right)-2{\left(a+1\right)}^{2}\hfill & \hfill \\ \hfill & \phantom{\rule{2em}{0ex}}-9\phantom{\rule{-0.17em}{0ex}}{\left(a-\frac{1}{3}\right)}^{2}\hfill \end{array}$
k)
$\begin{array}{ccc}\hfill & 2\left(2a-1\right)\left(2a+1\right)-{\left(a-2\right)}^{2}\hfill & \hfill \\ \hfill & \phantom{\rule{2em}{0ex}}-4\phantom{\rule{-0.17em}{0ex}}{\left(a+\frac{1}{2}\right)}^{2}\hfill \end{array}$
l)
$\begin{array}{ccc}\hfill & \phantom{\rule{-0.17em}{0ex}}{\left(a-\frac{1}{2}\right)}^{2}+\frac{1}{4}{\left(2a+1\right)}^{2}\hfill & \hfill \\ \hfill & \phantom{\rule{2em}{0ex}}-\frac{1}{2}\left(2a-1\right)\left(2a+1\right)\hfill \end{array}$

a)
${\left(x+3\right)}^{2}$
b)
${\left(a-5\right)}^{2}$
c)
$\left(x+3\right)\left(x-3\right)$
d)
$2{\left(x+4\right)}^{2}$
e)
$\phantom{\rule{-0.17em}{0ex}}{\left(x-\frac{1}{3}\right)}^{2}$
f)
${\left(x+2\right)}^{2}+{\left(x-3\right)}^{2}+\left(x+2\right)\left(x-2\right)$
g)
$\left(4-a\right)\left(4+a\right)-{\left(a+2\right)}^{2}+{\left(a-2\right)}^{2}$
h)
$-3{\left(2-x\right)}^{2}-5{\left(3x-4\right)}^{2}+2{x}^{2}$
i)
$-\left(4-x\right)\left(4+x\right)-{\left(2x-5\right)}^{2}-\phantom{\rule{-0.17em}{0ex}}{\left(x-\frac{1}{3}\right)}^{2}$
j)
$\left(2-a\right)\left(2+a\right)-2{\left(a+1\right)}^{2}-9\phantom{\rule{-0.17em}{0ex}}{\left(a-\frac{1}{3}\right)}^{2}$
k)
$2\left(2a-1\right)\left(2a+1\right)-{\left(a-2\right)}^{2}-4\phantom{\rule{-0.17em}{0ex}}{\left(a+\frac{1}{2}\right)}^{2}$
l)
$\phantom{\rule{-0.17em}{0ex}}{\left(a-\frac{1}{2}\right)}^{2}+\frac{1}{4}{\left(2a+1\right)}^{2}-\frac{1}{2}\left(2a-1\right)\left(2a+1\right)$

## 2.14)

Expand the parentheses and contract the resulting expressions:

a)
$\phantom{\rule{-0.17em}{0ex}}{\left(1+\frac{a}{2}\right)}^{2}$
b)
$\phantom{\rule{-0.17em}{0ex}}{\left(x+\frac{1}{x}\right)}^{2}$
c)
$\phantom{\rule{-0.17em}{0ex}}\left(\sqrt{x}+1\right)\phantom{\rule{-0.17em}{0ex}}\left(\sqrt{x}-1\right)$
d)
${\left(x-3\right)}^{2}+\left(x-2\right)\left(x+2\right)$

a)
$\phantom{\rule{-0.17em}{0ex}}{\left(1+\frac{a}{2}\right)}^{2}$
b)
$\phantom{\rule{-0.17em}{0ex}}{\left(x+\frac{1}{x}\right)}^{2}$
c)
$\phantom{\rule{-0.17em}{0ex}}\left(\sqrt{x}+1\right)\phantom{\rule{-0.17em}{0ex}}\left(\sqrt{x}-1\right)$
d)
${\left(x-3\right)}^{2}+\left(x-2\right)\left(x+2\right)$

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

a)
${x}^{2}-4x+4$
b)
${x}^{2}-36$
c)
$2{x}^{3}-4{x}^{2}+2x$
d)
$2{y}^{2}+20y+50$
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)
$\frac{25{a}^{2}-81}{5a-9}$
d)
$\frac{16{a}^{2}-36}{6}$
e)
$\frac{9{x}^{2}-4}{3x-2}$
f)
$\frac{9{a}^{2}-64}{6a+16}$

## 2.18)

Factorize the expressions:

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

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

## 2.19)

Simplify the expressions by using the algebraic identities:

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

a)
$\frac{{\left(x+2\right)}^{2}}{{x}^{2}-4}$
b)
$\frac{2{x}^{2}-8}{x-2}$
c)
$\frac{{x}^{2}-9}{2x+6}$
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)
$\frac{4{x}^{2}-8x+4}{2x-2}$
c)
$\frac{{x}^{2}+4x+4}{4-{x}^{2}}$
d)
$\frac{5-{x}^{2}}{\sqrt{5}-x}$
e)
$\frac{{\left(x+2y\right)}^{2}}{{x}^{2}-4{y}^{2}}$

## 2.21)

Solve without using a calculator:

Example 1

$\begin{array}{llll}\hfill 99×101& =\left(100-1\right)\left(100+1\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =10{0}^{2}-{1}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =10\phantom{\rule{0.17em}{0ex}}000-1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =9999\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

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

a)
$29×31$
b)
$18×22$
c)
$25×15$
d)
$103×97$

a)
$29×31$
b)
$18×22$
c)
$25×15$
d)
$103×97$

You can find the solutions here.