(a)

\begin{align} x^2 + 7x - 18 & = 0 \\ (x + 9)(x - 2) & = 0 \end{align} \begin{align} x + 9 & = 0 && \text{ or } & x - 2 & = 0 \\ x & = - 9 &&& x & = 2 \end{align}

(b)

\begin{align} 2x^2 + 5x - 7 & = 0 \\ (2x + 7)(x - 1) & = 0 \end{align} \begin{align} 2x + 7 & = 0 && \text{ or } & x - 1 & = 0 \\ 2x & = -7 &&& x & = 1 \\ x & = -{7 \over 2} \end{align}

(c)

\begin{align} 5y^2 - 28y + 15 & = 0 \\ (5y - 3)(y - 5) & = 0 \end{align} \begin{align} 5y - 3 & = 0 && \text{ or } & y - 5 & = 0 \\ 5y & = 3 &&& y & = 5 \\ y & = {3 \over 5} \end{align}

(d)

\begin{align} 4z^2 - 49 & = 0 \\ 4z^2 & = 49 \\ z^2 & = {49 \over 4} \\ z & = \pm \sqrt{49 \over 4} \\ z & = \pm {7 \over 2} \end{align}

(a)

\begin{align} (x + 1)^2 & = 9 \\ x + 1 & = \pm \sqrt{9} \\ x + 1 & = \pm 3 \end{align} \begin{align} x + 1 & = 3 && \text{ or } & x + 1 & = -3 \\ x & = 3 - 1 &&& x & = -3 - 1 \\ x & = 2 &&& x & = -4 \end{align}

(b)

\begin{align} (2y + 1)^2 & = 16 \\ 2y + 1 & = \pm \sqrt{16} \\ 2y + 1 & = \pm 4 \end{align} \begin{align} 2y + 1 & = 4 && \text{ or } & 2y + 1 & = -4 \\ 2y & = 4 - 1 &&& 2y & = -4 - 1 \\ 2y & = 3 &&& 2y & = -5 \\ y & = {3 \over 2} &&& y & = -{5 \over 2} \end{align}

(c)

\begin{align} (5h - 4)^2 & = 81 \\ 5h - 4 & = \pm \sqrt{81} \\ 5h - 4 & = \pm 9 \end{align} \begin{align} 5h - 4 & = 9 && \text{ or } & 5h - 4 & = -9 \\ 5h & = 9 + 4 &&& 5h & = -9 + 4 \\ 5h & = 13 &&& 5h & = -5 \\ h & = {13 \over 5} &&& h & = {-5 \over 5} \\ & &&& & = -1 \end{align}

(d)

\begin{align} (7 - 3k)^2 & = {9 \over 16} \\ 7 - 3k & = \pm \sqrt{9 \over 16} \\ 7 - 3k & = \pm {3 \over 4} \end{align} \begin{align} 7 - 3k & = {3 \over 4} && \text{ or } & 7 - 3k & = -{3 \over 4} \\ -3k & = {3 \over 4} - 7 &&& -3k & = -{3 \over 4} - 7 \\ -3k & = -{25 \over 4} &&& -3k & = -{31 \over 4} \\ k & = {-{25 \over 4} \over -3} &&& k & = {-{31 \over 4} \over -3} \\ k & = {25 \over 12} &&& k & = {31 \over 12} \end{align}

(e)

\begin{align} (m + 3)^2 & = 11 \\ m + 3 & = \pm \sqrt{11} \end{align} \begin{align} m + 3 & = \sqrt{11} && \text{ or } & m + 3 & = - \sqrt{11} \\ m & = \sqrt{11} - 3 &&& m & = - \sqrt{11} - 3 \\ m & = 0.31662 &&& m & = -6.3166 \\ m & \approx 0.32 \text{ (2 d.p.)} &&& m & = -6.32 \text{ (2 d.p.)} \end{align}

(f)

\begin{align} (2n - 3)^2 & = 23 \\ 2n - 3 & = \pm \sqrt{23} \end{align} \begin{align} 2n - 3 & = \sqrt{23} && \text{ or } & 2n - 3 & = - \sqrt{23} \\ 2n & = \sqrt{23} + 3 &&& 2n & = - \sqrt{23} + 3 \\ n & = {\sqrt{23} + 3 \over 2} &&& n & = {-\sqrt{23} + 3 \over 2} \\ n & = 3.8979 &&& n & = -0.89791 \\ n & \approx 3.90 \text{ (2 d.p.)} &&& n & \approx -0.90 \text{ (2 d.p.)} \end{align}

(g)

\begin{align} (5 - w)^2 & = 7 \\ 5 - w & = \pm \sqrt{7} \end{align} \begin{align} 5 - w & = \sqrt{7} && \text{ or } & 5 - w & = -\sqrt{7} \\ -w & = \sqrt{7} - 5 &&& -w & = - \sqrt{7} - 5 \\ w & = -\sqrt{7} + 5 &&& w & = \sqrt{7} + 5 \\ w & = 2.3542 &&& w & = 7.6457 \\ w & \approx 2.35 \text{ (2 d.p.)} &&& w & \approx 7.65 \text{ (2 d.p.)} \end{align}

(h)

\begin{align} \left({1 \over 2} - t\right)^2 & = 10 \\ {1 \over 2} - t & = \pm \sqrt{10} \end{align} \begin{align} {1 \over 2} - t & = \sqrt{10} && \text{ or } & {1 \over 2} - t & = - \sqrt{10} \\ -t & = \sqrt{10} - {1 \over 2} &&& -t & = -\sqrt{10} - {1 \over 2} \\ t & = - \sqrt{10} + {1 \over 2} &&& t & = \sqrt{10} + {1 \over 2} \\ t & = -2.6622 &&& t & = 3.6622 \\ t & \approx -2.66 \text{ (2 d.p.)} &&& t & \approx 3.66 \text{ (2 d.p.)} \end{align}

(a)

\begin{align} x^2 + 20x & = x^2 + 20x + \left(20 \over 2\right)^2 - \left(20 \over 2\right)^2 \\ & = x^2 + 20x + 10^2 - 10^2 \\ & = (x + 10)^2 - 10^2 \\ & = (x + 10)^2 - 100 \end{align}

(b)

\begin{align} x^2 - 15x & = x^2 - 15x + \left(15 \over 2\right)^2 - \left(15 \over 2\right)^2 \\ & = \left(x - {15 \over 2} \right)^2 - \left(15 \over 2\right)^2 \\ & = \left(x - {15 \over 2}\right)^2 - {225 \over 4} \end{align}

(c)

\begin{align} x^2 + {1 \over 2}x & = x^2 + {1 \over 2}x + \left({1 \over 2} \over 2\right)^2 - \left({1 \over 2} \over 2\right)^2 \\ & = x^2 + {1 \over 2}x + \left(1 \over 4\right)^2 - \left(1 \over 4\right)^2 \\ & = \left(x + {1 \over 4}\right)^2 - {1 \over 16} \end{align}

(d)

\begin{align} x^2 - {2 \over 9}x & = x^2 - {2 \over 9}x + \left({2 \over 9} \over 2\right)^2 - \left({2 \over 9} \over 2\right)^2 \\ & = x^2 - {2 \over 9}x + \left(1 \over 9\right)^2 - \left(1 \over 9\right)^2 \\ & = \left(x - {1 \over 9}\right)^2 - {1 \over 81} \end{align}

(e)

\begin{align} x^2 + 0.2x & = x^2 + 0.2x + \left(0.2 \over 2\right)^2 - \left(0.2 \over 2\right)^2 \\ & = x^2 + 0.2x + 0.1^2 - 0.1^2 \\ & = (x + 0.1)^2 - 0.1^2 \\ & = (x + 0.1)^2 - 0.01 \end{align}

(f)

\begin{align} x^2 - 1.4x & = x^2 - 1.4x + \left(1.4 \over 2\right)^2 - \left(1.4 \over 2\right)^2 \\ & = x^2 - 1.4x + 0.7^2 - 0.7^2 \\ & = (x - 0.7)^2 - 0.7^2 \\ & = (x - 0.7)^2 - 0.49 \end{align}

(g)

\begin{align} -x^2 - 10x & = -(x^2 + 10x) \\ & = - \left[ x^2 + 10x + \left(10 \over 2\right)^2 - \left(10 \over 2\right)^2 \right] \\ & = - ( x^2 + 10x + 5^2 - 5^2 ) \\ & = - [ (x + 5)^2 - 5^2 ] \\ & = - [ (x + 5)^2 - 25 ] \\ & = - (x + 5)^2 + 25 \end{align}

(h)

\begin{align} -x^2 + 11x & = -(x^2 - 11x) \\ & = - \left[ x^2 - 11x + \left(11 \over 2\right)^2 - \left(11 \over 2\right)^2 \right] \\ & = - \left[ \left(x - {11 \over 2}\right)^2 - \left(11 \over 2\right)^2 \right] \\ & = - \left[ \left(x - {11 \over 2}\right)^2 - {121 \over 4} \right] \\ & = - \left(x - {11 \over 2}\right)^2 + {121 \over 4} \end{align}

(a)

\begin{align} x^2 - 6x + 1 & = x^2 - 6x + \left(6 \over 2\right)^2 - \left(6 \over 2\right)^2 + 1 \\ & = x^2 - 6x + 3^2 - 3^2 + 1 \\ & = (x - 3)^2 - 3^2 + 1 \\ & = (x - 3)^2 - 8 \end{align}

(b)

\begin{align} x^2 + 3x - 2 & = x^2 + 3x + \left(3 \over 2\right)^2 - \left(3 \over 2\right)^2 - 2 \\ & = \left(x + {3 \over 2}\right)^2 - \left( 3 \over 2 \right)^2 - 2 \\ & = \left(x + {3 \over 2}\right)^2 - {17 \over 4} \end{align}

(c)

\begin{align} x^2 + 9x - 1.8 & = x^2 + 9x + \left(9 \over 2\right)^2 - \left(9 \over 2\right)^2 - 1.8 \\ & = x^2 + 9x + 4.5^2 - 4.5^2 - 1.8 \\ & = (x + 4.5)^2 - 4.5^2 - 1.8 \\ & = (x + 4.5)^2 - 22.05 \end{align}

(d)

\begin{align} x^2 - {2 \over 7}x + 7 & = x^2 - {2 \over 7}x + \left( {2 \over 7} \over 2\right)^2 - \left( {2 \over 7} \over 2\right)^2 + 7 \\ & = x^2 - {2 \over 7}x + \left(1 \over 7\right)^2 - \left(1 \over 7\right)^2 + 7 \\ & = \left(x - {1 \over 7}\right)^2 - \left(1 \over 7\right)^2 + 7 \\ & = \left(x - {1 \over 7}\right)^2 + {342 \over 49} \end{align}

(e)

\begin{align} -x^2 + 10x - 2 & = - (x^2 - 10x) - 2 \\ & = - \left[ x^2 - 10x + \left(10 \over 2\right)^2 - \left(10 \over 2\right)^2 \right] - 2 \\ & = - ( x^2 - 10x + 5^2 - 5^2 ) - 2 \\ & = - [ (x - 5)^2 - 5^2 ] - 2 \\ & = - [ (x - 5)^2 - 25 ] - 2 \\ & = - (x - 5)^2 + 25 - 2 \\ & = - (x - 5)^2 + 23 \end{align}

(f)

\begin{align} -x^2 + 13x - {13 \over 2} & = - (x^2 - 13x) - {13 \over 2} \\ & = - \left[ x^2 - 13x + \left(13 \over 2\right)^2 - \left(13 \over 2\right)^2 \right] - {13 \over 2} \\ & = - \left[ \left(x - {13 \over 2}\right)^2 - \left(13 \over 2\right)^2 \right] - {13 \over 2} \\ & = - \left[ \left(x - {13 \over 2}\right)^2 - {169 \over 4} \right] - {13 \over 2} \\ & = - \left(x - {13 \over 2}\right)^2 + {169 \over 4} - {13 \over 2} \\ & = - \left(x - {13 \over 2}\right)^2 + {143 \over 4} \end{align}

(g)

\begin{align} -x^2 - 9x - 20.25 & = -(x^2 + 9x) - 20.25 \\ & = - \left[ x^2 + 9x + \left(9 \over 2\right)^2 - \left(9 \over 2\right)^2 \right] - 20.25 \\ & = - (x^2 + 9x + 4.5^2 - 4.5^2) - 20.25 \\ & = - [ (x + 4.5)^2 - 4.5^2 ] - 20.25 \\ & = - [ (x + 4.5)^2 - 20.25 ] - 20.25 \\ & = - (x + 4.5)^2 + 20.25 - 20.25 \\ & = - (x + 4.5)^2 \end{align}

(h)

\begin{align} -x^2 - {3 \over 4}x + 3 & = - \left(x^2 + {3 \over 4}x \right) + 3 \\ & = - \left[ x^2 + {3 \over 4}x + \left({3 \over 4} \over 2\right)^2 - \left({3 \over 4} \over 2\right)^2 \right] + 3 \\ & = - \left[ x^2 + {3 \over 4}x + \left(3 \over 8\right)^2 - \left(3 \over 8\right)^2 \right] + 3 \\ & = - \left[ \left(x + {3 \over 8}\right)^2 - \left(3 \over 8\right)^2 \right] + 3 \\ & = - \left[ \left(x + {3 \over 8}\right)^2 - {9 \over 64} \right] + 3 \\ & = - \left(x + {3 \over 8}\right)^2 + {9 \over 64} + 3 \\ & = - \left(x + {3 \over 8}\right)^2 + {201 \over 64} \end{align}

(a)

\begin{align} x^2 + 2x - 5 & = 0 \\ x^2 + 2x + \left(2 \over 2\right)^2 - \left(2 \over 2\right)^2 - 5 & = 0 \\ x^2 + 2x + 1^2 - 1^2 - 5 & = 0 \\ (x + 1)^2 - 1^2 - 5 & = 0 \\ (x + 1)^2 - 6 & = 0 \\ (x + 1)^2 & = 6 \\ x + 1 & = \pm \sqrt{6} \end{align} \begin{align} x + 1 & = \sqrt{6} && \text{ or } & x + 1 & = - \sqrt{6} \\ x & = \sqrt{6} - 1 &&& x & = -\sqrt{6} - 1 \\ x & = 1.4494 &&& x & = -3.4494 \\ x & \approx 1.45 \text{ (2 d.p.)} &&& x & \approx -3.45 \text{ (2 d.p.)} \end{align}

(b)

\begin{align} y^2 - 12y + 9 & = 0 \\ y^2 - 12y + \left(12 \over 2\right)^2 - \left(12 \over 2\right)^2 + 9 & = 0 \\ y^2 - 12y + 6^2 - 6^2 + 9 & = 0 \\ (y - 6)^2 - 6^2 + 9 & = 0 \\ (y - 6)^2 - 27 & = 0 \\ (y - 6)^2 & = 27 \\ y - 6 & = \pm \sqrt{27} \end{align} \begin{align} y - 6 & = \sqrt{27} && \text{ or } & y - 6 & = -\sqrt{27} \\ y & = \sqrt{27} + 6 &&& y & = - \sqrt{27} + 6 \\ y & = 11.196 &&& y & = 0.80384 \\ y & \approx 11.20 \text{ (2 d.p.)} &&& y & \approx 0.80 \text{ (2 d.p.)} \end{align}

(c)

\begin{align} z^2 - 5z - 5 & = 0 \\ z^2 - 5z + \left(5 \over 2\right)^2 - \left(5 \over 2\right)^2 - 5 & = 0 \\ z^2 - 5z + 2.5^2 - 2.5^2 - 5 & = 0 \\ (z - 2.5)^2 - 2.5^2 - 5 & = 0 \\ (z - 2.5)^2 - 11.25 & = 0 \\ (z - 2.5)^2 & = 11.25 \\ z - 2.5 & = \pm \sqrt{11.25} \end{align} \begin{align} z - 2.5 & = \sqrt{11.25} && \text{ or } & z - 2.5 & = - \sqrt{11.25} \\ z & = \sqrt{11.25} + 2.5 &&& z & = - \sqrt{11.25} + 2.5 \\ z & = 5.8541 &&& z & = -0.85410 \\ z & \approx 5.85 \text{ (2 d.p.)} &&& z & \approx -0.85 \text{ (2 d.p.)} \end{align}

(d)

\begin{align} p^2 + {1 \over 4}p - 3 & = 0 \\ p^2 + {1 \over 4}p + \left({1 \over 4} \over 2\right)^2 - \left({1 \over 4} \over 2\right)^2 - 3 & = 0 \\ p^2 + {1 \over 4}p + \left(1 \over 8\right)^2 - \left(1 \over 8\right)^2 - 3 & = 0 \\ \left(p + {1 \over 8}\right)^2 - \left(1 \over 8\right)^2 - 3 & = 0 \\ \left(p + {1 \over 8}\right)^2 - {193 \over 64} & = 0 \\ \left(p + {1 \over 8}\right)^2 & = {193 \over 64} \\ p + {1 \over 8} & = \pm \sqrt{193 \over 64} \end{align} \begin{align} p + {1 \over 8} & = \sqrt{193 \over 64} && \text{ or } & p + {1 \over 8} & = - \sqrt{193 \over 64} \\ p & = \sqrt{193 \over 64} - {1 \over 8} &&& p & = - \sqrt{193 \over 64} - {1 \over 8} \\ p & = 1.6115 &&& p & = -1.8615 \\ p & \approx 1.61 \text{ (2 d.p.)} &&& p & \approx -1.86 \text{ (2 d.p.)} \end{align}

(e)

\begin{align} q^2 - {6 \over 7}q + {2 \over 49} & = 0 \\ q^2 - {6 \over 7}q + \left({6 \over 7} \over 2\right)^2 - \left({6 \over 7} \over 2\right)^2 + {2 \over 49} & = 0 \\ q^2 - {6 \over 7}q + \left(3 \over 7\right)^2 - \left(3 \over 7\right)^2 + {2 \over 49} & = 0 \\ \left(q - {3 \over 7}\right)^2 - \left(3 \over 7\right)^2 + {2 \over 49} & = 0 \\ \left(q - {3 \over 7}\right)^2 - {1 \over 7} & = 0 \\ \left(q - {3 \over 7}\right)^2 & = {1 \over 7} \\ q - {3 \over 7} & = \pm \sqrt{1 \over 7} \end{align} \begin{align} q - {3 \over 7} & = \sqrt{1 \over 7} && \text{ or } & q - {3 \over 7} & = - \sqrt{1 \over 7} \\ q & = \sqrt{1 \over 7} + {3 \over 7} &&& q & = - \sqrt{1 \over 7} + {3 \over 7} \\ q & = 0.80653 &&& q & = 0.05060 \\ q & \approx 0.81 \text{ (2 d.p.)} &&& q & \approx 0.05 \text{ (2 d.p.)} \end{align}

(f)

\begin{align} r^2 + 0.6r - 1 & = 0 \\ r^2 + 0.6r + \left(0.6 \over 2 \right)^2 - \left(0.6 \over 2\right)^2 - 1 & = 0 \\ r^2 + 0.6r + 0.3^2 - 0.3^2 - 1 & = 0 \\ (r + 0.3)^2 - 0.3^2 - 1 & = 0 \\ (r + 0.3)^2 - 1.09 & = 0 \\ (r + 0.3)^2 & = 1.09 \\ r + 0.3 & = \pm \sqrt{1.09} \end{align} \begin{align} r + 0.3 & = \sqrt{1.09} && \text{ or } & r + 0.3 & = -\sqrt{1.09} \\ r & = \sqrt{1.09} - 0.3 &&& r & = - \sqrt{1.09} - 0.3 \\ r & = 0.74403 &&& r & = -1.3440 \\ r & \approx 0.74 \text{ (2 d.p.)} &&& r & \approx -1.34 \text{ (2 d.p.)} \end{align}

(i)

\begin{align} x^2 + 17x - 30 & = x^2 + 17x + \left(17 \over 2\right)^2 - \left(17 \over 2\right)^2 - 30 \\ & = \left(x + {17 \over 2}\right)^2 - \left(17 \over 2\right)^2 - 30 \\ & = \left(x + {17 \over 2}\right)^2 - {409 \over 4} \end{align}

(ii)

\begin{align} x^2 + 17x - 30 & = 0 \\ \\ \text{From (i), } x^2 + 17x - 30 & = \left(x + {17 \over 2}\right)^2 - {409 \over 4} \\ \\ \\ \left(x + {17 \over 2}\right)^2 - {409 \over 4} & = 0 \\ \left(x + {17 \over 2}\right)^2 & = {409 \over 4} \\ x + {17 \over 2} & = \pm \sqrt{409 \over 4} \end{align} \begin{align} x + {17 \over 2} & = \sqrt{409 \over 4} && \text{ or } & x + {17 \over 2} & = - \sqrt{409 \over 4} \\ x & = \sqrt{409 \over 4} - {17 \over 2} &&& x & = - \sqrt{409 \over 4} - {17 \over 2} \\ x & = 1.6118 &&& x & = -18.611 \\ x & \approx 1.6 \text{ (1 d.p.)} &&& x & \approx -18.6 \text{ (1 d.p.)} \end{align}

(a)

\begin{align} a(a + 4) & = 3a + 1 \\ a^2 + 4a & = 3a + 1 \\ a^2 + 4a - 3a - 1 & = 0 \\ a^2 + a - 1 & = 0 \\ a^2 + a + \left(1 \over 2\right)^2 - \left(1 \over 2\right)^2 - 1 & = 0\\ a^2 + a + 0.5^2 - 0.5^2 - 1 & = 0 \\ (a + 0.5)^2 - 0.5^2 - 1 & = 0 \\ (a + 0.5)^2 - 1.25 & = 0 \\ (a + 0.5)^2 & = 1.25 \\ a + 0.5 & = \pm \sqrt{1.25} \end{align} \begin{align} a + 0.5 & = \sqrt{1.25} && \text{ or } & a + 0.5 & = -\sqrt{1.25} \\ a & = \sqrt{1.25} - 0.5 &&& a & = -\sqrt{1.25} - 0.5 \\ a & = 0.61803 &&& a & = -1.6180 \\ a & \approx 0.618 &&& a & \approx -1.62 \end{align}

(b)

\begin{align} (b + 1)^2 & = 7b \\ (b)^2 + 2(b)(1) + (1)^2 & = 7b \phantom{000000} [(a + b)^2 = a^2 + 2ab + b^2] \\ b^2 + 2b + 1 & = 7b \\ b^2 + 2b - 7b + 1 & = 0 \\ b^2 - 5b + 1 & = 0 \\ b^2 - 5b + \left(5 \over 2\right)^2 - \left(5 \over 2\right)^2 + 1 & = 0 \\ b^2 - 5b + 2.5^2 - 2.5^2 + 1 & = 0 \\ (b - 2.5)^2 - 2.5^2 + 1 & = 0 \\ (b - 2.5)^2 - 5.25 & = 0 \\ (b - 2.5)^2 & = 5.25 \\ b - 2.5 & = \pm \sqrt{5.25} \end{align} \begin{align} b - 2.5 & = \sqrt{5.25} && \text{ or } & b - 2.5 & = - \sqrt{5.25} \\ b & = \sqrt{5.25} + 2.5 &&& b & = - \sqrt{5.25} + 2.5 \\ b & = 4.7912 &&& b & = 0.20871 \\ b & \approx 4.79 &&& b & \approx 0.209 \end{align}

(c)

\begin{align} (c - 2)(c + 5) & = c \\ c^2 + 5c - 2c - 10 & = c \\ c^2 + 3c - 10 & = c \\ c^2 + 3c - c - 10 & = 0 \\ c^2 + 2c - 10 & = 0 \\ c^2 + 2c + \left(2 \over 2\right)^2 - \left(2 \over 2\right)^2 - 10 & = 0 \\ c^2 + 2c + 1^2 - 1^2 - 10 & = 0 \\ (c + 1)^2 - 1^2 - 10 & = 0 \\ (c + 1)^2 - 11 & = 0 \\ (c + 1)^2 & = 11 \\ c + 1 & = \pm \sqrt{11} \end{align} \begin{align} c + 1 & = \sqrt{11} && \text{ or } & c + 1 & = - \sqrt{11} \\ c & = \sqrt{11} - 1 &&& c & = - \sqrt{11} - 1 \\ c & = 2.3166 &&& c & = -4.3166 \\ c & \approx 2.32 &&& c & \approx -4.32 \end{align}

(d)

\begin{align} d(d - 4) & = 2(d + 7) \\ d^2 - 4d & = 2d + 14 \\ d^2 - 4d - 2d - 14 & = 0 \\ d^2 - 6d - 14 & = 0 \\ d^2 - 6d + \left(6 \over 2\right)^2 - \left(6 \over 2\right)^2 - 14 & = 0 \\ d^2 - 6d + 3^2 - 3^2 - 14 & = 0 \\ (d - 3)^2 - 3^2 - 14 & = 0 \\ (d - 3)^2 - 23 & = 0 \\ (d - 3)^2 & = 23 \\ d - 3 & = \pm \sqrt{23} \end{align} \begin{align} d - 3 & = \sqrt{23} && \text{ or } & d - 3 & = - \sqrt{23} \\ d & = \sqrt{23} + 3 &&& d & = - \sqrt{23} + 3 \\ d & = 7.7958 &&& d & = -1.7958 \\ d & \approx 7.80 &&& d & \approx -1.80 \end{align}

\begin{align} y^2 - ay - 6 & = 0 \\ y^2 - ay + \left(a \over 2\right)^2 - \left(a \over 2\right)^2 - 6 & = 0 \\ \left( y - {a \over 2} \right)^2 - \left(a \over 2\right)^2 - 6 & = 0 \\ \left( y - {a \over 2} \right)^2 - {a^2 \over 4} - 6 & = 0 \\ \left( y - {a \over 2} \right)^2 & = {a^2 \over 4} + 6 \\ \left( y - {a \over 2} \right)^2 & = {a^2 \over 4} + {6 \over 1} \\ \left( y - {a \over 2} \right)^2 & = {a^2 \over 4} + {24 \over 4} \\ \left( y - {a \over 2} \right)^2 & = {a^2 + 24 \over 4} \\ y - {a \over 2} & = \pm \sqrt{ a^2 + 24 \over 4 } \\ y - {a \over 2} & = { \pm \sqrt{a^2 + 24} \over \sqrt{4} } \\ y - {a \over 2} & = { \pm \sqrt{a^2 + 24} \over 2 } \\ y & = { \pm \sqrt{a^2 + 24} \over 2} + {a \over 2} \\ y & = { \pm \sqrt{a^2 + 24} + a \over 2 } \\ y & = {a \pm \sqrt{a^2 + 24} \over 2} \end{align}