$$ \hspace*{2 mm}\mathrm{1.1\kern3mmx^2 + 4x − 3\kern2mm\ } $$
Add the square of half of
the coefficient of x and then
subtract it again :
$$ \hspace*{2 mm}\mathrm{1.1\kern3mmx^2 + 4x − 3\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x^2 + 4x + (\frac{1}{2} \times \frac{4}{1})^2 − (\frac{1}{2} \times \frac{4}{1})^2 − 3)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x^2 + 4x + (\frac{2}{1})^2 − 4 − 3)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((x + 2)^2 − 4 − 3)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((x + 2)^2 − 7)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x + 2)^2 − 7\kern2mm\ } $$
The minimum value is −7,
because if x = -2, then (x + 2)
2 = 0
and for all other real values of x,
(x + 2) > 0, so that
(x + 2)
2 − 7 > − 7
[ Q 1.1 ]
$$ \hspace*{2 mm}\mathrm{1.2\kern3mmx^2 − 3x + 2\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x^2 − 3x + (\frac{1}{2} \times \frac{−3}{1})^2 − (\frac{1}{2} \times \frac{−3}{1})^2 + 2)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x^2 − 3x + (\frac{−3}{1})^2 − (\frac{9}{4}) + 2)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((x − \frac{3}{2})^2 − (\frac{9}{4}) + 2)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((x − \frac{3}{2})^2 − (\frac{1}{4}))\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x − \frac{3}{2})^2 − \frac{1}{4}\kern2mm\ } $$
The minimum value is −¼,
because if x =
3⁄
2, then (x −
3⁄
2)
2 = 0
and for all other values of x
(x −
3⁄
2)
2 > 0 so that
(x −
3⁄
2)
2 − ¼ > − ¼.
[ Q 1.2 ]
$$ \hspace*{2 mm}\mathrm{1.3\kern3mmx^2 + 4x − 3\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x^2 + 4x + (\frac{1}{2} \times \frac{4}{1})^2 − (\frac{1}{2} \times \frac{4}{1})^2 − 3)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((x + 2)^2 − 4 − 3)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((x + 2)^2 − 7)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x + 2)^2 − 7)\kern2mm\ } $$
The minimum value is −7
[ Q 1.3 ]
$$ \hspace*{2 mm}\mathrm{1.4\kern3mmx^2 − 5x + 6\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x^2 − 5x + (\frac{1}{2} \times \frac{−5}{1})^2 − (\frac{1}{2} \times \frac{−5}{1})^2 + 6)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((x − \frac{5}{2})^2 − \frac{25}{4} + 6\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x − \frac{5}{2})^2 − ¼\kern2mm\ } $$
The minimum value is −¼
[ Q 1.4 ]
$$ \hspace*{2 mm}\mathrm{1.5\kern3mma^2 − 3a + 1\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (a^2 − 3a + (\frac{1}{2} \times \frac{−3}{1})^2 − (\frac{1}{2} \times \frac{−3}{1})^2 + 1)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (a − \frac{3}{2})^2 − \frac{9}{4} + 1\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (a − \frac{3}{2})^2 − \frac{5}{4}\kern2mm\ } $$
The minimum value is −
5⁄
4
[ Q 1.5 ]
$$ \hspace*{2 mm}\mathrm{1.6\kern3mm2a^2 − 3a + 1\kern2mm\ } $$
Take the coefficient of
x
2 as common factor.
$$ \hspace*{2 mm}\mathrm{1.6\kern3mm2a^2 − 3a + 1\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2(a^2 − \frac{3}{2}a + \frac{1}{2})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2(a^2 − \frac{3}{2}a + (\frac{1}{2} \times \frac{−3}{2})^2 − (\frac{1}{2} \times \frac{−3}{2})^2 + \frac{1}{2})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2(a^2 − \frac{3}{2}a + (\frac{−3}{4})^2 − (\frac{−3}{4})^2 + \frac{1}{2})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2((a − \frac{3}{4})^2 − \frac{9}{16} + \frac{1}{2})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2((a − \frac{3}{4})^2 − \frac{1}{16})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2(a − \frac{3}{4})^2 − \frac{1}{8}\kern2mm\ } $$
The minimum value is −
1⁄
8
[ Q 1.6 ]
$$ \hspace*{2 mm}\mathrm{1.7\kern3mm3p^2 − 4p − 2\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 3(p^2 − \frac{4}{3}p − \frac{2}{3})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 3(p^2 − \frac{4}{3}p + (\frac{1}{2} \times \frac{−4}{3})^2 − (\frac{1}{2} \times \frac{−4}{3})^2 − \frac{2}{3})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 3(p^2 − \frac{4}{3}p + (\frac{−2}{3})^2 − \frac{4}{9} − \frac{2}{3})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 3((p − \frac{2}{3})^2 − \frac{4}{9} − \frac{6}{9})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 3((p − \frac{2}{3})^2 − \frac{10}{9})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 3(p − \frac{2}{3})^2 − \frac{10}{3}\kern2mm\ } $$
The minimum value is −
10⁄
3
[ Q 1.7 ]
$$ \hspace*{2 mm}\mathrm{1.8\kern3mm8q^2 + 3q − 5\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8(q^2 + \frac{3}{8}q − \frac{5}{8})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8(q^2 + \frac{3}{8}q + (\frac{1}{2} \times \frac{3}{8})^2 − (\frac{1}{2} \times \frac{3}{8})^2 − \frac{5}{8})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8(q^2 + \frac{3}{8}q + (\frac{3}{16})^2 − \frac{9}{256} − \frac{5}{8})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8((q + \frac{3}{16})^2 − \frac{169}{256})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8(q + \frac{3}{16})^2 − \frac{169}{32}\kern2mm\ } $$
The minimum value is −
169⁄
32
[ Q 1.8 ]
$$ \hspace*{2 mm}\mathrm{1.9\kern3mm7x^2 − 2x − 5\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 7(x^2 − \frac{2}{7}x − \frac{5}{7})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 7(x^2 − \frac{2}{7}x + (\frac{1}{2} \times \frac{−2}{7})^2 − (\frac{1}{2} \times \frac{−2}{7})^2 − \frac{5}{7})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 7((x − \frac{1}{7})^2 − \frac{1}{49} − \frac{5}{7})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 7((x − \frac{1}{7})^2 − \frac{36}{49})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 7(x − \frac{1}{7})^2 − \frac{36}{7})\kern2mm\ } $$
The minimum value is −
36⁄
7
[ Q 1.9 ]
$$ \hspace*{2 mm}\mathrm{1.10\kern3mm8x^2 − 2x + 3\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8(x^2 − \frac{2}{8}x + \frac{3}{8})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8(x^2 − \frac{1}{4}x + (\frac{1}{2} \times \frac{−2}{8})^2 − (\frac{1}{2} \times \frac{−2}{7})^2 − \frac{3}{8})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8((x − \frac{1}{8})^2 − \frac{1}{64} − \frac{3}{8})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8((x − \frac{1}{8})^2 − \frac{25}{64})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8(x − \frac{1}{8})^2 − \frac{25}{8})\kern2mm\ } $$
The minimum value is −
25⁄
8
[ Q 1.10 ]
$$ \hspace*{2 mm}\mathrm{1.11\kern3mm−y^2 + 5y − 6\kern2mm\ } $$
Take −1 as common factor
OR place everything in a
negative bracket and
remember to change ALL the
signs inside the bracket.
$$ \hspace*{2 mm}\mathrm{1.11\kern3mm−y^2 + 5y − 6\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −(y^2 − 5y + 6)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −(y^2 − 5y + (\frac{1}{2} \times \frac{−5}{1})^2 − (\frac{1}{2} \times \frac{−5}{1})^2 + 6)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −((y − \frac{5}{2})^2 − \frac{25}{4} + 6)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −((y − \frac{5}{2})^2 − \frac{1}{4})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −(y − \frac{5}{2})^2 + \frac{1}{4}\kern2mm\ } $$
The minimum value is ¼, because
if y =
5⁄
2, then (y −
5⁄
2)
2 = 0 and for
all other real values of y
−(y −
5⁄
2)
2 < 0 so that
−(y −
5⁄
2)
2 + ¼ < ¼.
[ Q 1.11 ]
$$ \hspace*{2 mm}\mathrm{1.12\kern3mmx^2 + 3x + 1\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x^2 + 3x + (\frac{1}{2} \times \frac{3}{1})^2 − (\frac{1}{2} \times \frac{3}{1})^2 + 1)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((x + \frac{3}{2})^2 − \frac{9}{4} + 1)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x + \frac{3}{2})^2 − \frac{5}{4}\kern2mm\ } $$
The minimum value is −
5⁄
4
[ Q 1.12 ]
$$ \hspace*{2 mm}\mathrm{1.13\kern3mm−y^2 − 5y − 8\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −(y^2 + 5y + 8)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −(y^2 + 5y + (\frac{1}{2} \times \frac{5}{1})^2 − (\frac{1}{2} \times \frac{5}{1})^2 + 8)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −((y + \frac{5}{2})^2 − \frac{25}{4} + 8)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −((y + \frac{5}{2})^2 − \frac{1}{4})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −(y + \frac{5}{2})^2 + \frac{1}{4}\kern2mm\ } $$
The maximum value is ¼,
because if y = −
5⁄
2, then
(y +
5⁄
2)
2 = 0 for all other real
values of y (y +
5⁄
2)
2 < 0
so that −(y +
5⁄
2)
2 + ¼ < ¼.
[ Q 1.13 ]
$$ \hspace*{2 mm}\mathrm{1.14\kern3mm−3a^2 − 2a − 1\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −3(a^2 + \frac{2}{3}a + \frac{1}{3})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −3(a^2 + \frac{2}{3}a + (\frac{1}{2} \times \frac{2}{3})^2 − (\frac{1}{2} \times \frac{2}{3})^2 + \frac{1}{3})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −3((a + \frac{1}{3})^2 − \frac{1}{9} + \frac{1}{3})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −3((a + \frac{1}{3})^2 + \frac{2}{9})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −3(a + \frac{1}{3})^2 − \frac{2}{3}\kern2mm\ } $$
The maximum value is −
2⁄
3
[ Q 1.14 ]
$$ \hspace*{2 mm}\mathrm{1.15\kern3mm2a^2 − 3a + 2\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2(a^2 − \frac{3}{2}a + 1)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2(a^2 − \frac{3}{2}a + (\frac{1}{2} \times \frac{−3}{2})^2 − (\frac{1}{2} \times \frac{−3}{2})^2 + 1)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2((a − \frac{3}{4})^2 − \frac{9}{16} + 1)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2(a − \frac{3}{4})^2 + \frac{7}{8}\kern2mm\ } $$
The minimum value is
7⁄
8
[ Q 1.15 ]
$$ \hspace*{2 mm}\mathrm{1.16\kern3mm−4p^2 + p − 1\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −4(p^2 − \frac{1}{4}p + \frac{1}{4})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −4(p^2 − \frac{1}{4}p + (\frac{1}{2} \times \frac{−1}{4})^2 − (\frac{1}{2} \times \frac{−1}{4})^2 + \frac{1}{4})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −4((p − \frac{1}{8})^2 − \frac{1}{64} + \frac{1}{4})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −4(p − \frac{1}{8})^2 + \frac{15}{16}\kern2mm\ } $$
The maximum value is
15⁄
16
[ Q 1.16 ]
$$ \hspace*{2 mm}\mathrm{1.17\kern3mm−3q^2 + 2q − 8\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −3(q^2 − \frac{2}{3}q + \frac{8}{3})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −3(q^2 − \frac{2}{3}q + (\frac{1}{2} \times \frac{−2}{3})^2 − (\frac{1}{2} \times \frac{−2}{3})^2 + \frac{8}{3})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −3((q − \frac{1}{3})^2 − \frac{1}{9} + \frac{8}{3})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −3(q − \frac{1}{3})^2 − \frac{23}{3}\kern2mm\ } $$
The maximum value is −
23⁄
3
[ Q 1.17 ]
$$ \hspace*{2 mm}\mathrm{1.19\kern3mm−5x^2 − 3x − 7\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −5(x^2 − \frac{3}{5}x + \frac{7}{5})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −5(x^2 − \frac{2}{3}x + (\frac{1}{2} \times \frac{−3}{5})^2 − (\frac{1}{2} \times \frac{−3}{5})^2 + \frac{7}{5})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −5((x − \frac{3}{10})^2 − \frac{9}{100} + \frac{7}{5})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ −5(x − \frac{3}{10})^2 − \frac{131}{20}\kern2mm\ } $$
The maximum value is −
131⁄
20
[ Q 1.18 ]
$$ \hspace*{2 mm}\mathrm{2.1\kern3mmx^2 + 2x\kern2mm\ } $$
First complete the sqaure and
then determine what has to
be added.
$$ \hspace*{2 mm}\mathrm{2.1\kern3mmx^2 + 2x\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x^2 + 2x + (\frac{1}{2} \times \frac{2}{1})^2 − (\frac{1}{2} \times \frac{2}{1})^2)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((x + 1)^2 − 1)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (x + 1)^2 − 1\kern2mm\ } $$
1 must be added because
x
2 + x + 1 = (x
2 + x) + 1
= (x + 1)
2 - 1 + 1
= (x + 1)
2
which is a perfect square.
[ Q 2.1 ]
$$ \hspace*{2 mm}\mathrm{2.2\kern3mmy^2 − y\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (y^2 − y + (\frac{1}{2} \times \frac{−1}{1})^2 − (\frac{1}{2} \times \frac{−1}{1})^2)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((y − \frac{1}{2})^2 − \frac{1}{4})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (y − \frac{1}{2})^2 − \frac{1}{4}\kern2mm\ } $$
1⁄
4 must be added because
y
2 − y +
1⁄
4 = (y
2 − y) +
1⁄
4
= (y − ½)
2 − ¼ + ¼
= (y − ½)
2
which is a perfect square.
[ Q 2.2 ]
$$ \hspace*{2 mm}\mathrm{2.3\kern3mma^2 + 2a + 4\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (a^2 + 2a + (\frac{1}{2} \times \frac{2}{1})^2 − (\frac{1}{2} \times \frac{2}{1})^2 + 4)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((a + 1)^2 − 1 + 4)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (a + 1)^2 + 3\kern2mm\ } $$
−3 must be added
[ Q 2.3 ]
$$ \hspace*{2 mm}\mathrm{2.4\kern3mmp^2 + 2p − 3\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (p^2 + 2p + (\frac{1}{2} \times \frac{2}{1})^2 − (\frac{1}{2} \times \frac{2}{1})^2 − 3)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((p + 1)^2 − 1 − 3)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (p + 1)^2 − 4\kern2mm\ } $$
4 must be added
[ Q 2.4 ]
$$ \hspace*{2 mm}\mathrm{2.5\kern3mmq^2 − 3q − 5\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (q^2 − 3q + (\frac{1}{2} \times \frac{−3}{1})^2 − (\frac{1}{2} \times \frac{−3}{1})^2 − 5)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((q − \frac{3}{2})^2 − \frac{9}{4} − 5)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (q − \frac{3}{2})^2 − \frac{29}{4}\kern2mm\ } $$
29⁄
4 must be added.
[ Q 2.5 ]
$$ \hspace*{2 mm}\mathrm{2.6\kern3mma^2 + 2a − 5\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (a^2 + 2a + (\frac{1}{2} \times \frac{2}{1})^2 − (\frac{1}{2} \times \frac{2}{1})^2 − 5)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ ((a + 1)^2 − 1 − 5)\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ (a + 1)^2 − 6\kern2mm\ } $$
6 must be added.
[ Q 2.6 ]
$$ \hspace*{2 mm}\mathrm{2.7\kern3mm2p^2 + p − 1\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2(p^2 + \frac{1}{2}p + (\frac{1}{2} \times \frac{1}{2})^2 − (\frac{1}{2} \times \frac{1}{2})^2 − \frac{1}{2})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2((p + \frac{1}{4})^2 − \frac{1}{16} − \frac{1}{2})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2((p + \frac{1}{4})^2 − \frac{9}{16})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 2(p + \frac{1}{4})^2 − \frac{9}{8}\kern2mm\ } $$
9⁄
8 must be added.
[ Q 2.7 ]
$$ \hspace*{2 mm}\mathrm{2.8\kern3mm3q^2 + 2q + 4\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 3(q^2 + \frac{2}{3}q + (\frac{1}{2} \times \frac{2}{3})^2 − (\frac{1}{2} \times \frac{2}{3})^2 + \frac{4}{3})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 3((q + \frac{1}{3})^2 − \frac{1}{9} + \frac{4}{3})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 3((q + \frac{1}{3})^2 + \frac{11}{9})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 3(q + \frac{1}{3})^2 + \frac{11}{3}\kern2mm\ } $$
−
11⁄
3 must be added.
[ Q 2.8 ]
$$ \hspace*{2 mm}\mathrm{2.9\kern3mm5x^2 − 3x + 2\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 5(x^2 − \frac{3}{5}x + (\frac{1}{2} \times \frac{−3}{5})^2 − (\frac{1}{2} \times \frac{−3}{5})^2 + \frac{2}{5})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 5((x − \frac{3}{10})^2 − \frac{9}{100} + \frac{2}{5})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 5((x − \frac{3}{10})^2 + \frac{31}{100})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 5(x − \frac{3}{10})^2 + \frac{31}{20}\kern2mm\ } $$
−
31⁄
20 must be added.
[ Q 2.9 ]
$$ \hspace*{2 mm}\mathrm{2.10\kern3mm5x^2 − 3x + 2\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 5(x^2 − \frac{3}{5}x + (\frac{1}{2} \times \frac{−3}{5})^2 − (\frac{1}{2} \times \frac{−3}{5})^2 + \frac{2}{5})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 5((x − \frac{3}{10})^2 − \frac{9}{100} + \frac{2}{5})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 5((x − \frac{3}{10})^2 + \frac{31}{100})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 5(x − \frac{3}{10})^2 + \frac{31}{20}\kern2mm\ } $$
−
31⁄
20 must be added.
[ Q 2.10 ]
$$ \hspace*{2 mm}\mathrm{2.11\kern3mm7y^2 + 5y + 4\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 7(y^2 + \frac{5}{7}x + (\frac{1}{2} \times \frac{5}{7})^2 − (\frac{1}{2} \times \frac{5}{7})^2 + \frac{4}{7})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 7((y + \frac{5}{14})^2 − \frac{25}{196} + \frac{4}{7})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 7((y + \frac{5}{14})^2 + \frac{87}{196})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 7(y + \frac{5}{14})^2 + \frac{87}{28}\kern2mm\ } $$
−
87⁄
28 must be added.
[ Q 2.11 ]
$$ \hspace*{2 mm}\mathrm{2.12\kern3mm8z^2 − 3z − 7\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8(z^2 − \frac{3}{8}z + (\frac{1}{2} \times \frac{−3}{8})^2 − (\frac{1}{2} \times \frac{−3}{8})^2 − \frac{7}{8})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8((z − \frac{3}{16})^2 − \frac{9}{256} − \frac{7}{8})\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8((z − \frac{3}{10})^2 − \frac{233}{256}\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{=\ 8(z − \frac{3}{10})^2 − \frac{233}{32}\kern2mm\ } $$
233⁄
32 must be added.
[ Q 2.12 ]