MATHEMATICS
Grade 11
MORE EXERCISES
Solving quadratic equations : answers
  
  
Answers  1
    
$$ \hspace*{2 mm}\mathrm{1.1\kern3mmx^2 + 4x + 3 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x + 1)(x + 3) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x + 1) = 0\ \ \ OR\ \ \ (x + 3) = 0\kern2mm\ } $$
$$ \hspace*{14 mm}\mathrm{x = −1\ \ \ OR\ \ \ x = −3\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Test : x^2 + 4x + 3 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{x = −1 : (−1)^2 + 4(−1) + 3 \kern2mm\ } $$ $$ \hspace*{30 mm}\mathrm{ = 1 −4 + 3\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{ = 0\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{x = −1\ is\ a\ root.\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{Toets : x^2 + 4x + 3 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{x = −3 : (−3)^2 + 4(−3) + 3 \kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{ = 9 −12 + 3\kern2mm\ } $$
$$ \hspace*{30 mm}\mathrm{ = 0\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{x = −3\ is\ a\ root\kern2mm\ } $$
                                              [ Q 1.1 ]
    
$$ \hspace*{2 mm}\mathrm{1.2\kern3mmx^2 + 7x + 10 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x + 2)(x + 5) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x + 2) = 0\ \ \ OR\ \ \ (x + 5) = 0\kern2mm\ } $$
$$ \hspace*{14 mm}\mathrm{x = −2\ \ \ OR\ \ \ x = −5\kern2mm\ } $$
                                              [ Q 1.2 ]
    
$$ \hspace*{2 mm}\mathrm{1.3\kern3mmy^2 + 3y + 2 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(y + 1)(y + 2) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(y + 1) = 0\ \ \ OR\ \ \ (y + 2) = 0\kern2mm\ } $$
$$ \hspace*{14 mm}\mathrm{y = −1\ \ \ OR\ \ \ y = −2\kern2mm\ } $$
                                              [ Q 1.3 ]
    
$$ \hspace*{2 mm}\mathrm{1.4\kern3mmy^2 − 10y + 21 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(y − 3)(y − 7) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(y − 3) = 0\ \ \ OR\ \ \ (y − 7) = 0\kern2mm\ } $$
$$ \hspace*{14 mm}\mathrm{y = 3\ \ \ OR\ \ \ y = 7\kern2mm\ } $$
                                              [ Q 1.4 ]
    
$$ \hspace*{2 mm}\mathrm{1.5\kern3mma^2 + a − 2 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(a − 1)(a + 2) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(a − 1) = 0\ \ \ OR\ \ \ (a + 2) = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{a = 1\ \ \ OR\ \kern9mm\ a = −2\kern2mm\ } $$
                                              [ Q 1.5 ]
    
$$ \hspace*{2 mm}\mathrm{1.6\kern3mmb^2 − b − 6 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(b − 3)(b + 2) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(b − 3) = 0\ \ \ OR\ \ \ (b + 2) = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{b = 3\ \ \ OR\ \kern9mm\ b = −2\kern2mm\ } $$
                                              [ Q 1.6 ]
    
$$ \hspace*{2 mm}\mathrm{1.7\kern3mmp^2 − 3p − 4 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(p − 4)(p + 1) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(p − 4) = 0\ \ \ OR\ \ \ (p + 1) = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{p = 4\ \ \ OR\ \kern9mm\ p = −1\kern2mm\ } $$
                                              [ Q 1.7 ]
    
$$ \hspace*{2 mm}\mathrm{1.8\kern3mmq^2 + 2q − 15 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(q − 3)(q + 5) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(q − 3) = 0\ \ \ OR\ \ \ (q + 5) = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{q = 3\ \ \ OR\ \kern9mm\ q = −5\kern2mm\ } $$
                                              [ Q 1.8 ]
    
$$ \hspace*{2 mm}\mathrm{1.9\kern3mmx^2 − 4x − 21 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x − 7)(x + 3) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x − 7) = 0\ \ \ OR\ \ \ (x + 3) = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{x = 7\ \ \ OR\ \kern9mm\ x = −3\kern2mm\ } $$
                                              [ Q 1.9 ]
    
$$ \hspace*{2 mm}\mathrm{1.10\kern3mm2x^2 + 3x + 1 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x + 1)(2x + 1) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x + 1) = 0\ \ \ OR\ \ \ (2x + 1) = 0\kern2mm\ } $$
$$ \hspace*{14 mm}\mathrm{x = −1\ \ \ OR\ \kern9mm\ x = −\frac{1}{2}\kern2mm\ } $$
                                              [ Q 1.10 ]
    
$$ \hspace*{2 mm}\mathrm{1.11\kern3mm3y^2 − 11y + 10 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(y − 2)(3y − 5) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(y − 2) = 0\ \ \ OR\ \ \ (3y − 5) = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{y = 2\ \ \ OR\ \kern9mm\ x = −\frac{5}{3}\kern2mm\ } $$
                                              [ Q 1.11 ]
    
$$ \hspace*{2 mm}\mathrm{1.12\kern3mm6z^2 + 13z + 6 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(2z + 3)(3z + 2) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(2z + 3) = 0\ \ \ OR\ \ \ (3z + 2) = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{z = −\frac{3}{2}\ \ \ OR\ \kern9mm\ z = −\frac{2}{3}\kern2mm\ } $$
                                              [ Q 1.12 ]
    
$$ \hspace*{2 mm}\mathrm{1.13\kern3mm15z^2 + 13z − 6 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(3z − 1)(5z + 6) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(3z − 3) = 0\ \ \ OR\ \ \ (5z + 6) = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{z = \frac{1}{3}\ \ \ OR\ \kern9mm\ z = −\frac{6}{5}\kern2mm\ } $$
                                              [ Q 1.13 ]
    
$$ \hspace*{2 mm}\mathrm{1.14\kern3mm6a^2 − 7a − 20 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(3a + 4)(2a − 5) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(3a + 4) = 0\ \ \ OR\ \ \ (2a − 5) = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{a = −\frac{4}{3}\ \ \ OR\ \kern9mm\ a = \frac{5}{2}\kern2mm\ } $$
                                              [ Q 1.14 ]
    
$$ \hspace*{2 mm}\mathrm{1.15\kern3mm35b^2 − 22b − 24 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(5b − 6)(7b + 4) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(5b − 6) = 0\ \ \ OR\ \ \ (7b + 5) = 0\kern2mm\ } $$
$$ \hspace*{17 mm}\mathrm{a = \frac{6}{5}\ \ \ OR\ \kern9mm\ b = −\frac{5}{7}\kern2mm\ } $$
                                              [ Q 1.15 ]
    
$$ \hspace*{2 mm}\mathrm{1.16\kern3mm−y^2 + 3y − 2 = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{y^2 − 3y + 2 = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{(y − 1)(y − 2) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(y − 1) = 0\ \ \ OR\ \ \ (y − 2) = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{y = 1\ \ \ OR\ \kern9mm\ y = 2\kern2mm\ } $$

$$ \hspace*{8 mm}\mathrm{Test : −y^2 + 3y − 2 = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{y = 1 : −(1)^2 + 3(1) − 2\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= −1 + 3 − 2\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{1\ is\ a\ root\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{y = 2 : −(2)^2 + 3(2) − 2\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= −4 + 6 − 2\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{= 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{2\ is\ a\ root\kern2mm\ } $$
                                              [ Q 1.16 ]
    
$$ \hspace*{2 mm}\mathrm{1.17\kern3mm−p^2 − 4p + 5 = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{p^2 + 4p − 5 = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{(p + 5)(p − 1) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(p + 5) = 0\ \ \ OR\ \ \ (p − 1) = 0\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{p = −5\ \ \ OR\ \kern9mm\ p = 1\kern2mm\ } $$

                                              [ Q 1.17 ]
    
$$ \hspace*{2 mm}\mathrm{1.18\kern3mm−3y^2 + 11y − 10 = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{3y^2 − 11y + 10 = 0\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{(3y − 5)(y − 2) = 0\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(3y − 5) = 0\ \ \ OR\ \ \ (y − 2) = 0\kern2mm\ } $$
$$ \hspace*{17 mm}\mathrm{y = \frac{5}{3}\ \ \ OR\ \kern9mm\ y = 2\kern2mm\ } $$

                                              [ Q 1.18 ]


  
Answers  2
    
$$ \hspace*{2 mm}\mathrm{2.1\kern3mmx^2 + x − 3 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = 1\ \kern4mm\ b = 1\ \kern4mm\ c = −3\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{x = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(1) \pm \sqrt{(1)^2 − 4(1)(−3)}}{2(1)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−1 \pm \sqrt{13}}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{x =\ \frac{−1 + \sqrt{13}}{2}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ 1,30\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x =\ \frac{−1 − \sqrt{13}}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ −2,30\kern2mm\ } $$
                                              [ Q 2.1 ]
    
$$ \hspace*{2 mm}\mathrm{2.2\kern3mmy^2 − y − 5 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = 1\ \kern4mm\ b = −1\ \kern4mm\ c = −5\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{y = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(−1) \pm \sqrt{(−1)^2 − 4(1)(−5)}}{2(1)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−1 \pm \sqrt{21}}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−1 \pm 4,582575...}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{y =\ \frac{−1 + 4,582575..}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ 1,79\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y =\ \frac{−1 − 4,582575..}{2}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ −2,79\kern2mm\ } $$
                                              [ Q 2.2 ]


    
$$ \hspace*{2 mm}\mathrm{2.3\kern3mmz^2 + 4z + 1 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = 1\ \kern4mm\ b = 4\ \kern4mm\ c = 1\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{z = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(4) \pm \sqrt{(4)^2 − 4(1)(1)}}{2(1)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−4 \pm \sqrt{12}}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−4 \pm 3,46410...}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{z =\ \frac{−4 + 3,46410..}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ −0,27\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{z =\ \frac{−4 −3,4641 ..}{2}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ −3,73\kern2mm\ } $$
                                              [ Q 2.3 ]


    
$$ \hspace*{2 mm}\mathrm{2.4\kern3mma^2 − 5a + 2 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = 1\ \kern4mm\ b = −5\ \kern4mm\ c = 2\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(−5) \pm \sqrt{(−5)^2 − 4(1)(2)}}{2(1)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{5 \pm \sqrt{17}}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{5 \pm 4,1231...}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{a =\ \frac{5 + 4,1231..}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ 4,56\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{a =\ \frac{5 −4,1231 ..}{2}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ 0,44\kern2mm\ } $$
                                              [ Q 2.4 ]
    
$$ \hspace*{2 mm}\mathrm{2.5\kern3mmb^2 + 7b − 4 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = 1\ \kern4mm\ b = 7\ \kern4mm\ c = −4\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{b = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(7) \pm \sqrt{(7)^2 − 4(1)(−4)}}{2(1)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{5 \pm \sqrt{65}}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{5 \pm 8,0623...}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{b =\ \frac{5 + 8,0623..}{2}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ 6,53\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{b =\ \frac{5 −8,0623 ..}{2}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ −1,53\kern2mm\ } $$
                                              [ Q 2.5 ]
    
$$ \hspace*{2 mm}\mathrm{2.6\kern3mm2a^2 − a − 1 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = 2 \kern4mm\ b = −1\ \kern4mm\ c = −1\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(−1) \pm \sqrt{(−1)^2 − 4(2)(−1)}}{2(2)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{1 \pm \sqrt{9}}{4}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{1 \pm 3}{4}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{a =\ \frac{1 + 3}{4}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ 1\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{a =\ \frac{1 −3}{4}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ −\frac{1}{2}\kern2mm\ } $$
                                              [ Q 2.6 ]
    
$$ \hspace*{2 mm}\mathrm{2.7\kern3mm3b^2 − 4b − 5 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = 3 \kern4mm\ b = −4\ \kern4mm\ c = −5\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{b = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(−4) \pm \sqrt{(−4)^2 − 4(3)(−5)}}{2(3)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{4 \pm \sqrt{76}}{6}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{4 \pm 8,7178...}{6}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{b =\ \frac{4 + 8,7178}{6}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ 2,12\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{b =\ \frac{4 −8,7178}{6}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ −0,79\kern2mm\ } $$
                                              [ Q 2.7 ]
    
$$ \hspace*{2 mm}\mathrm{2.8\kern3mm4p^2 − p − 7 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = 4 \kern4mm\ b = −1\ \kern4mm\ c = −7\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{p = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(−1) \pm \sqrt{(−1)^2 − 4(4)(−7)}}{2(4)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{1 \pm \sqrt{113}}{8}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{1 \pm 10,6301...}{8}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{p =\ \frac{1 + 10,6301..}{8}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ 1,45\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{p =\ \frac{1 −10,6301..}{8}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ −1,20\kern2mm\ } $$
                                              [ Q 2.8 ]
    
$$ \hspace*{2 mm}\mathrm{2.9\kern3mm5x^2 − 3x − 2 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = 5 \kern4mm\ b = −3\ \kern4mm\ c = −2\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{x = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(−3) \pm \sqrt{(−3)^2 − 4(5)(−2)}}{2(5)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{3 \pm \sqrt{49}}{10}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{1 \pm 7}{10}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{x =\ \frac{1 + 7}{10}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ 0,80\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{x =\ \frac{1 −7}{10}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ −0,60\kern2mm\ } $$
                                              [ Q 2.9 ]
    
$$ \hspace*{2 mm}\mathrm{2.10\kern3mm−2y^2 + 8y = 7\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{−2y^2 + 8y − 7 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = −2 \kern4mm\ b = 8\ \kern4mm\ c = −7\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{y = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(8) \pm \sqrt{(8)^2 − 4(−2)(−7)}}{2(−2)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−8 \pm \sqrt{8}}{−4}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−8 \pm 2,8284..}{−4}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{y =\ \frac{−8 + 2,8284..}{−4}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ −1,29\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{y =\ \frac{−8 − 2,8284..}{−4}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ 2,71\kern2mm\ } $$
                                              [ Q 2.10 ]
    
$$ \hspace*{2 mm}\mathrm{2.11\kern3mm−3z^2 − 4z = −9\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{−3z^2 − 4z + 9 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = −3 \kern4mm\ b = −4\ \kern4mm\ c = 9\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{z = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(−4) \pm \sqrt{(−4)^2 − 4(−3)(9)}}{2(−3)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{4 \pm \sqrt{124}}{−6}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{4 \pm 11,1355..}{−6}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{z =\ \frac{4 + 11,1355..}{−6}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ −2,52\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{z =\ \frac{4 − 11,1355..}{−6}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ 1,19\kern2mm\ } $$
                                              [ Q 2.11 ]
    
$$ \hspace*{2 mm}\mathrm{2.12\kern3mm−4a^2 + 1 = 3a\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{−4a^2 − 3a + 1 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = −4 \kern4mm\ b = −3\ \kern4mm\ c = 1\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{a = \frac{−b \pm \sqrt{b^2 − 4ac}}{2a}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{−(−3) \pm \sqrt{(−3)^2 − 4(−4)(1)}}{2(−4)}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{3 \pm \sqrt{25}}{−8}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{=\ \frac{3 \pm 5}{−8}\kern2mm\ } $$

$$ \hspace*{13 mm}\mathrm{a =\ \frac{3 + 5}{−8}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ −1,00\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{OR\kern2mm\ } $$
$$ \hspace*{13 mm}\mathrm{a =\ \frac{3 − 5}{−8}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{=\ 0,25\kern2mm\ } $$
                                              [ Q 2.12 ]
  
Answers  3
    
$$ \hspace*{2 mm}\mathrm{3.1\kern3mmq^2 − 2q − 3 = 0\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{q^2 − 2q + (\frac{1}{2} \times \frac{−2}{1})^2\ − (\frac{1}{2} \times \frac{−2}{1})^2\ − 3 = 0\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{(q − 1)^2 − 1 − 3 = 0\kern2mm\ } $$
$$ \hspace*{12 mm}\mathrm{(q − 1)^2 − 4 = 0\kern2mm\ } $$
$$ \hspace*{18 mm}\mathrm{(q − 1)^2 = 4\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{(q − 1) = \pm \sqrt{4}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{(q − 1) = \pm 2\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(q − 1) = 2\ \kern2mm\ OR\ \kern2mm\ (q − 1) = −2\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{q = 3\ \kern2mm\ OR\ \kern2mm\ q = −1\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{\bold{\large{OR}}\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{3.1\kern3mmq^2 − 2q − 3 = 0\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{q^2 − 2q = 3\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{q^2 − 2q + (\frac{1}{2} \times \frac{−2}{1})^2\ =\ (\frac{1}{2} \times \frac{−2}{1})^2\ + 3\kern2mm\ } $$

$$ \hspace*{19 mm}\mathrm{(q − 1)^2 = 1 + 3\kern2mm\ } $$
$$ \hspace*{19 mm}\mathrm{(q − 1)^2 = 4\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{(q − 1) = \pm \sqrt{4}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{(q − 1) = \pm 2\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(q − 1) = 2\ \kern2mm\ OR\ \kern2mm\ (q − 1) = −2\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{q = 3\ \kern2mm\ OR\ \kern2mm\ q = −1\kern2mm\ } $$
                                              [ Q 3.1 ]
    
$$ \hspace*{2 mm}\mathrm{3.2\kern3mmx^2 + 3x − 4 = 0\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{x^2 + 3x + (\frac{1}{2} \times \frac{3}{1})^2\ − (\frac{1}{2} \times \frac{3}{1})^2\ − 4 = 0\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{(x + \frac{3}{2})^2 − \frac{9}{4} − 4 = 0\kern2mm\ } $$
$$ \hspace*{9 mm}\mathrm{(x + \frac{3}{2})^2 − \frac{25}{4} = 0\kern2mm\ } $$
$$ \hspace*{17 mm}\mathrm{(x + \frac{3}{2})^2 = \frac{25}{4}\kern2mm\ } $$
$$ \hspace*{18 mm}\mathrm{(x + \frac{3}{2}) = \sqrt{\frac{25}{4}}\kern2mm\ } $$
$$ \hspace*{18 mm}\mathrm{(x + \frac{3}{2}) = \pm 2,5\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{(x + \frac{3}{2}) = 2,5\ \kern2mm\ OR\ \kern2mm\ (x + \frac{3}{2}) = −2,5\kern2mm\ } $$

$$ \hspace*{18 mm}\mathrm{x = 1\ \kern2mm\ OR\ \kern2mm\ x = −4\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{\bold{\large{OR}}\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{3.2\kern3mmx^2 + 3x − 4 = 0\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{x^2 + 3x = 4\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{x^2 + 3x + (\frac{1}{2} \times \frac{3}{1})^2\ =\ (\frac{1}{2} \times \frac{3}{1})^2\ + 4\kern2mm\ } $$

$$ \hspace*{19 mm}\mathrm{(x + \frac{3}{2})^2 = \frac{9}{4} + 4\kern2mm\ } $$

$$ \hspace*{19 mm}\mathrm{(x + \frac{3}{2})^2 = \frac{25}{4}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{(x + \frac{3}{2}) = \pm \sqrt{\frac{25}{4}}\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(x + \frac{3}{2}) = \pm 2.5\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{(x + \frac{3}{2}) = 2,5\ \kern2mm\ OR\ \kern2mm\ (x + \frac{3}{2}) = −2,5\kern2mm\ } $$
$$ \hspace*{18 mm}\mathrm{x = 1\ \kern2mm\ OR\ \kern2mm\ x = −4\kern2mm\ } $$
                                              [ Q 3.2 ]
    
$$ \hspace*{2 mm}\mathrm{3.3\kern3mmy^2 − 2y − 8 = 0\kern2mm\ } $$ $$ \hspace*{15 mm}\mathrm{y^2 − 2y = 8\kern2mm\ } $$

$$ \hspace*{6 mm}\mathrm{y^2 − 2y + (\frac{1}{2} \times \frac{−2}{1})^2\ =\ (\frac{1}{2} \times \frac{−2}{1})^2\ + 8\kern2mm\ } $$

$$ \hspace*{19 mm}\mathrm{(y − 1)^2 = 1 + 8\kern2mm\ } $$ $$ \hspace*{19 mm}\mathrm{(y − 1)^2 = 9\kern2mm\ } $$ $$ \hspace*{20 mm}\mathrm{(y − 1) = \pm \sqrt{9}\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(y − 1) = \pm 3\kern2mm\ } $$ $$ \hspace*{8 mm}\mathrm{(y − 1) = 3\ \kern2mm\ OR\ \kern2mm\ (y − 1) = −3\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{y = 4\ \kern2mm\ OR\ \kern2mm\ y = −2\kern2mm\ } $$
                                              [ Q 3.3 ]
    
$$ \hspace*{2 mm}\mathrm{3.4\kern3mm3z^2 + z − 2 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{z^2 + \frac{1}{3}z − \frac{2}{3} = 0\kern2mm\ } $$

$$ \hspace*{6 mm}\mathrm{z^2 + \frac{1}{3}z + (\frac{1}{2} \times \frac{1}{3})^2\ − (\frac{1}{2} \times \frac{1}{3})^2\ − \frac{2}{3} = 0\kern2mm\ } $$

$$ \hspace*{6 mm}\mathrm{(z + \frac{1}{6})^2 − \frac{1}{36} − \frac{2}{3} = 0\kern2mm\ } $$

$$ \hspace*{9 mm}\mathrm{(x + \frac{1}{6})^2 − \frac{25}{36} = 0\kern2mm\ } $$

$$ \hspace*{17 mm}\mathrm{(x + \frac{1}{6})^2 = \frac{25}{36}\kern2mm\ } $$

$$ \hspace*{18 mm}\mathrm{(x + \frac{1}{6}) = \sqrt{\frac{25}{36}}\kern2mm\ } $$

$$ \hspace*{18 mm}\mathrm{(x + \frac{1}{6}) = \pm \frac{5}{6}\kern2mm\ } $$

$$ \hspace*{6 mm}\mathrm{(x + \frac{1}{6}) = \frac{5}{6}\ \kern2mm\ OR\ \kern2mm\ (x + \frac{1}{6}) = −\frac{5}{6}\kern2mm\ } $$

$$ \hspace*{15 mm}\mathrm{x = \frac{4}{6}\ \kern2mm\ OR\ \kern2mm\ x = −1\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{x = 0,67\ \kern2mm\ OR\ \kern2mm\ x = −1\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{\bold{\large{OR}}\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{3.4\kern3mm3z^2 + z − 2 = 0\kern2mm\ } $$
$$ \hspace*{15 mm}\mathrm{z^2 + \frac{1}{3}z = \frac{2}{3}\kern2mm\ } $$

$$ \hspace*{6 mm}\mathrm{z^2 + \frac{1}{3}z + (\frac{1}{2} \times \frac{1}{3})^2\ =\ (\frac{1}{2} \times \frac{1}{3})^2\ + \frac{2}{3}\kern2mm\ } $$

$$ \hspace*{19 mm}\mathrm{(z + \frac{1}{6})^2 = \frac{1}{36} + \frac{2}{3}\kern2mm\ } $$

$$ \hspace*{19 mm}\mathrm{(z + \frac{1}{2})^2 = \frac{25}{36}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{(z + \frac{1}{6}) = \pm \sqrt{\frac{25}{36}}\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(z + \frac{1}{6}) = \pm \frac{5}{6}\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(z + \frac{1}{6}) = \frac{5}{6}\ \kern2mm\ OR\ \kern2mm\ (z + \frac{1}{6}) = −\frac{5}{6}\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{z = \frac{4}{6}\ \kern2mm\ OR\ \kern2mm\ x = −1\kern2mm\ } $$
$$ \hspace*{11 mm}\mathrm{z = 0,67\ \kern2mm\ OR\ \kern2mm\ x = −1\kern2mm\ } $$
                                              [ Q 3.4 ]
    
$$ \hspace*{2 mm}\mathrm{3.5\kern3mm2a^2 − 3a − 4 = 0\kern2mm\ } $$
$$ \hspace*{18mm}\mathrm{a^2 − \frac{3}{2}a = 2\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{a^2 − \frac{3}{2}a + (\frac{1}{2} \times \frac{−3}{2})^2\ =\ (\frac{1}{2} \times \frac{3}{2})^2\ + 2\kern2mm\ } $$

$$ \hspace*{19 mm}\mathrm{(a − \frac{3}{4})^2 = \frac{9}{16} + 2\kern2mm\ } $$

$$ \hspace*{19 mm}\mathrm{(a − \frac{3}{4})^2 = \frac{41}{16}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{(a − \frac{3}{4}) = \pm \sqrt{\frac{41}{16}}\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(a − \frac{3}{4}) = \pm 1,6008..\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{(a − \frac{3}{4}) = 1,6008\ \kern2mm\ OR\ \kern2mm\ (a − \frac{3}{4}) = −1,6008\kern2mm\ } $$
$$ \hspace*{18 mm}\mathrm{a = 2,35\ \kern2mm\ OR\ \kern2mm\ a = 0,85\kern2mm\ } $$
                                              [ Q 3.5 ]
    
$$ \hspace*{2 mm}\mathrm{3.6\kern3mmb^2 + b + 6 = 0\kern2mm\ } $$
$$ \hspace*{18mm}\mathrm{b^2 + b = − 6\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{b^2 + b + (\frac{1}{2} \times \frac{1}{1})^2\ =\ (\frac{1}{2} \times \frac{1}{1})^2\ − 6\kern2mm\ } $$

$$ \hspace*{19 mm}\mathrm{(b + \frac{1}{2})^2 = \frac{1}{4} − 6\kern2mm\ } $$

$$ \hspace*{19 mm}\mathrm{(b + \frac{1}{2})^2 = −\frac{23}{4}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{(b + \frac{1}{2}) = −(\pm \sqrt{\frac{23}{4}})\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(b + \frac{1}{2}) = −(\pm 2,3979..)\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{(b + \frac{1}{2}) = −2,3979\ \kern2mm\ OR\ \kern2mm\ (b + \frac{1}{2}) = 2,3979\kern2mm\ } $$
$$ \hspace*{21 mm}\mathrm{b = 2,90\ \kern2mm\ OR\ \kern2mm\ b = 1,90\kern2mm\ } $$
                                              [ Q 3.6 ]
    
$$ \hspace*{2 mm}\mathrm{3.7\kern3mm−p^2 − 3p + 7 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{p^2 + 3p − 7 = 0\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{p^2 + 3p + (\frac{1}{2} \times \frac{3}{1})^2 − (\frac{1}{2} \times \frac{3}{1})^2 − 7 = 0\kern2mm\ } $$

$$ \hspace*{6 mm}\mathrm{(p + \frac{3}{2})^2 − \frac{9}{4} − 7 = 0\kern2mm\ } $$

$$ \hspace*{12 mm}\mathrm{(p + \frac{3}{2})^2 − \frac{37}{4} = 0\kern2mm\ } $$

$$ \hspace*{18 mm}\mathrm{(p + \frac{3}{2})^2 = \frac{37}{4}\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(p + \frac{3}{2}) = \pm \sqrt{\frac{37}{4}}\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(p + \frac{3}{2}) = \pm 3,0414\kern2mm\ } $$

$$ \hspace*{8 mm}\mathrm{(p + \frac{3}{2}) = 3,0414\ \kern2mm\ OR\ \kern2mm\ (p + \frac{3}{2}) = −3,0414\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{p = 1,54\ \kern2mm\ OR\ \kern2mm\ p = −4,54\kern2mm\ } $$
$$ \hspace*{28 mm}\mathrm{\bold{\large{OR}}\kern2mm\ } $$
$$ \hspace*{2 mm}\mathrm{3.7\kern3mm−p^2 − 3p + 7 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{p^2 + 3p = 7\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{p^2 + 3p + (\frac{1}{2} \times \frac{3}{1})^2 = (\frac{1}{2} \times \frac{3}{1})^2 + 7\kern2mm\ } $$

$$ \hspace*{21 mm}\mathrm{(p + \frac{3}{2})^2 = \frac{9}{4} + 7\kern2mm\ } $$

$$ \hspace*{21 mm}\mathrm{(p + \frac{3}{2})^2 = \frac{37}{4}\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(p + \frac{3}{2}) = \pm \sqrt{\frac{37}{4}}\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(p + \frac{3}{2}) = \pm 3,0414\kern2mm\ } $$

$$ \hspace*{8 mm}\mathrm{(p + \frac{3}{2}) = 3,0414\ \kern2mm\ OR\ \kern2mm\ (p + \frac{3}{2}) = −3,0414\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{p = 1,54\ \kern2mm\ OR\ \kern2mm\ p = −4,54\kern2mm\ } $$
                                              [ Q 3.7 ]
    
$$ \hspace*{2 mm}\mathrm{3.8\kern3mm−2q^2 + 5q − 6 = 0\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{q^2 − \frac{5}{2}q = − 3\kern2mm\ } $$
$$ \hspace*{6 mm}\mathrm{q^2 − \frac{5}{2}q + (\frac{1}{2} \times \frac{−5}{2})^2 = (\frac{1}{2} \times \frac{−5}{2})^2 − 3\kern2mm\ } $$

$$ \hspace*{21 mm}\mathrm{(q − \frac{5}{4})^2 = \frac{25}{16} − 3\kern2mm\ } $$

$$ \hspace*{21 mm}\mathrm{(q − \frac{5}{4})^2 = −\frac{23}{16}\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(q − \frac{5}{4}) = −(\pm \sqrt{\frac{23}{16}})\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(q − \frac{5}{4}) = \mp 1,1990\kern2mm\ } $$

$$ \hspace*{8 mm}\mathrm{(q − \frac{5}{4}) = −1,1990\ \kern2mm\ OR\ \kern2mm\ (q − \frac{5}{4}) = 1,1990\kern2mm\ } $$

$$ \hspace*{16 mm}\mathrm{q = 0,05\ \kern2mm\ OR\ \kern2mm\ q = −2,45\kern2mm\ } $$
                                              [ Q 3.8 ]
    
$$ \hspace*{2 mm}\mathrm{3.9\kern3mm−3x^2 + 4x = 7\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{x^2 − \frac{4}{3}x = −\frac{7}{3}\kern2mm\ } $$

$$ \hspace*{6 mm}\mathrm{x^2 − \frac{4}{3}x + (\frac{1}{2} \times \frac{−4}{3})^2 = (\frac{1}{2} \times \frac{−4}{3})^2 − \frac{7}{3}\kern2mm\ } $$

$$ \hspace*{21 mm}\mathrm{(x − \frac{2}{3})^2 = \frac{4}{9} − \frac{7}{3}\kern2mm\ } $$

$$ \hspace*{21 mm}\mathrm{(x − \frac{2}{3})^2 = −\frac{17}{9}\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(x − \frac{2}{3}) = −(\pm \sqrt{\frac{17}{9}})\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{(x − \frac{2}{3}) = \mp 1,3744\kern2mm\ } $$

$$ \hspace*{8 mm}\mathrm{(x − \frac{2}{3}) = −1,3744\ \kern2mm\ OR\ \kern2mm\ (x − \frac{2}{3}) = 1,3744\kern2mm\ } $$

$$ \hspace*{20 mm}\mathrm{x = −0,71\ \kern2mm\ OR\ \kern2mm\ x = 2,04\kern2mm\ } $$
                                              [ Q 3.9 ]
    
$$ \hspace*{2 mm}\mathrm{3.10\kern3mm−x^2 + 2x + 8 = 0\kern2mm\ } $$
$$ \hspace*{21 mm}\mathrm{x^2 − 2x = 8\kern2mm\ } $$
$$ \hspace*{10 mm}\mathrm{x^2 − 2x + (\frac{1}{2} \times \frac{−2}{1})^2 = (\frac{1}{2} \times \frac{−2}{1})^2 + 8\frac{7}{3}\kern2mm\ } $$

$$ \hspace*{21 mm}\mathrm{(x − 1)^2 = 1 + 8\kern2mm\ } $$
$$ \hspace*{21 mm}\mathrm{(x − 1)^2 = 9\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{(x − 1) = \pm \sqrt{9}\kern2mm\ } $$
$$ \hspace*{20 mm}\mathrm{(x − 1) = \pm 3\kern2mm\ } $$
$$ \hspace*{8 mm}\mathrm{(x − 1) = 3\ \kern2mm\ OR\ \kern2mm\ (x − 1) = −3\kern2mm\ } $$
$$ \hspace*{16 mm}\mathrm{x = 4\ \kern2mm\ OR\ \kern2mm\ x = −2\kern2mm\ } $$
                                              [ Q 3.10 ]