Ejercicio 267 – Solucionario Baldor

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CAPITULO XXXIII

Ecuaciones de segundo grado

Ejercicio 267

Resolver las siguientes ecuaciones aplicando la fórmula particular:

x = - \frac{m}{2} \pm \sqrt{\frac{m^{2}}{4} - n}

$\begin{matrix} x^{2} - 3 x + 2 & = 0 \\ m = - 3 \\ n = 2 \\ x = \frac{3}{2} \pm \sqrt{\frac{{(- 3)}^{2}}{4} - 2} \\ x = \frac{3}{2} \pm \sqrt{\frac{9}{4} - 2} \\ x = \frac{3}{2} \pm \sqrt{\frac{9 - 8}{4}} \\ x = \frac{3}{2} \pm \sqrt{\frac{1}{4}} \\ x = \frac{3}{2} \pm \frac{1}{2} \\ x_{1} = \frac{3}{2} + \frac{1}{2} = \frac{}{2} \\ x_{1} = 2 \\ x_{2} = \frac{3}{2} - \frac{1}{2} = \frac{2}{2} \\ x_{2} = 1 \end{matrix}$
$\begin{matrix} x^{2} - 2 x - 15 & = 0 \\ m = - 2 \\ n = - 15 \\ x = \frac{2}{2} \pm \sqrt{\frac{{(- 2)}^{2}}{4} - (- 15)} \\ x = 1 \pm \sqrt{\frac{4}{4} + 15} \\ x = 1 \pm \sqrt{1 + 15} \\ x = 1 \pm \sqrt{16} \\ x = 1 \pm 4 \\ x_{1} = 1 + 4 \\ x_{1} = 5 \\ x_{2} = 1 - 4 \\ x_{2} = - 3 \end{matrix}$
$\begin{matrix} x^{2} & = 19 x - 88 \\ x^{2} - 19 x + 88 & = 0 \\ m = - 19 \\ n = 88 \\ x = \frac{19}{2} \pm \sqrt{\frac{{(- 19)}^{2}}{4} - 88} \\ x = \frac{19}{2} \pm \sqrt{\frac{361}{4} - 88} \\ x = \frac{19}{2} \pm \sqrt{\frac{361 - 352}{4}} \\ x = \frac{19}{2} \pm \sqrt{\frac{9}{4}} \\ x = \frac{19}{2} \pm \frac{3}{2} \\ x_{1} = \frac{19}{2} + \frac{3}{2} = \frac{}{2} \\ x_{1} = 11 \\ x_{2} = \frac{19}{2} - \frac{3}{2} = \frac{}{2} \\ x_{2} = 8 \end{matrix}$
$\begin{matrix} x^{2} + 4 x & = 285 \\ x^{2} + 4 x - 285 & = 0 \\ m = 4 \\ n = - 285 \\ x = - \frac{}{2} \pm \sqrt{\frac{4^{2}}{4} - (- 285)} \\ x = - 2 \pm \sqrt{4 + 285} \\ x = - 2 \pm \sqrt{289} \\ x = - 2 \pm 17 \\ x_{1} = - 2 + 17 \\ x_{1} = 15 \\ x_{2} = - 2 - 17 \\ x_{2} = - 19 \end{matrix}$
$\begin{matrix} 5 x (x - 1) - 2 (2 x^{2} - 7 x) & = - 8 \\ 5 x^{2} - 5 x - 4 x^{2} + 14 x + 8 & = 0 \\ x^{2} + 9 x + 8 & = 0 \\ m = 9 \\ n = 8 \\ x = - \frac{9}{2} \pm \sqrt{\frac{9^{2}}{4} - 8} \\ x = - \frac{9}{2} \pm \sqrt{\frac{81}{4} - 8} \\ x = - \frac{9}{2} \pm \sqrt{\frac{81 - 32}{4}} \\ x = - \frac{9}{2} \pm \sqrt{\frac{49}{4}} \\ x = - \frac{9}{2} \pm \frac{7}{2} \\ x_{1} = - \frac{9}{2} + \frac{7}{2} = \frac{- 9 + 7}{2} = - \frac{2}{2} \\ x_{1} = - 1 \\ x_{2} = - \frac{9}{2} - \frac{7}{2} = \frac{- 9 - 7}{2} = - \frac{}{2} \\ x_{2} = - 8 \end{matrix}$
$\begin{matrix} x^{2} - (7 x + 6) & = x + 59 \\ x^{2} - 7 x - 6 - x - 59 & = 0 \\ x^{2} - 8 x - 65 & = 0 \\ m = - 8 \\ n = - 65 \\ x = \frac{}{2} \pm \sqrt{\frac{8^{2}}{4} - (- 65)} \\ x = 4 \pm \sqrt{\frac{}{4} + 65} \\ x = 4 \pm \sqrt{16 + 65} \\ x = 4 \pm \sqrt{81} \\ x = 4 \pm 9 \\ x_{1} = 4 + 9 \\ x_{1} = 13 \\ x_{2} = 4 - 9 \\ x_{2} = - 5 \end{matrix}$
$\begin{matrix} {(x - 1)}^{2} + 11 x + 199 & = 3 x^{2} - {(x - 2)}^{2} \\ x^{2} - 2 x + 1 + 11 x + 199 & = 3 x^{2} - (x^{2} - 4 x + 4) \\ x^{2} + 9 x + 200 & = 3 x^{2} - x^{2} + 4 x - 4 \\ x^{2} + 9 x + 200 - 2 x^{2} - 4 x + 4 & = 0 \\ - x^{2} + 5 x + 204 & = 0 \\ x^{2} - 5 x - 204 & = 0 \\ m = - 5 \\ n = - 204 \\ x = \frac{5}{2} \pm \sqrt{\frac{{(- 5)}^{2}}{4} - (- 204)} \\ x = \frac{5}{2} \pm \sqrt{\frac{25}{4} + 204} \\ x = \frac{5}{2} \pm \sqrt{\frac{25 + 816}{4}} \\ x = \frac{5}{2} \pm \sqrt{\frac{841}{4}} \\ x = \frac{5}{2} \pm \frac{29}{2} \\ x_{1} = \frac{5}{2} + \frac{29}{2} = \frac{5 + 29}{2} = \frac{}{2} \\ x_{1} = 17 \\ x_{2} = \frac{5}{2} - \frac{29}{2} = \frac{5 - 29}{2} = - \frac{}{2} \\ x_{2} = - 12 \end{matrix}$
$\begin{matrix} (x - 2) (x + 2) - 7 (x - 1) & = 21 \\ x^{2} - 4 - 7 x + 7 & = 21 \\ x^{2} - 7 x + 3 - 21 & = 0 \\ x^{2} - 7 x - 18 & = 0 \\ m = - 7 \\ n = - 17 \\ x = \frac{7}{2} \pm \sqrt{\frac{{(- 7)}^{2}}{4} - (- 18)} \\ x = \frac{7}{2} \pm \sqrt{\frac{49}{4} + 18} \\ x = \frac{7}{2} \pm \sqrt{\frac{49 + 72}{4}} \\ x = \frac{7}{2} \pm \sqrt{\frac{121}{4}} \\ x = \frac{7}{2} \pm \frac{11}{2} \\ x_{1} = \frac{7}{2} + \frac{11}{2} = \frac{7 + 11}{2} = \frac{}{2} \\ x_{1} = 9 \\ x_{2} = \frac{7}{2} - \frac{11}{2} = \frac{7 - 11}{2} = - \frac{}{2} \\ x_{2} = - 2 \end{matrix}$
$\begin{matrix} 2 x^{2} - (x - 2) (x + 5) & = 7 (x + 3) \\ 2 x^{2} - (x^{2} + 3 x - 10) & = 7 x + 21 \\ 2 x^{2} - x^{2} - 3 x + 10 - 7 x - 21 & = 0 \\ x^{2} - 10 x - 11 & = 0 \\ m = - 10 \\ n = - 11 \\ x = \frac{}{2} \pm \sqrt{\frac{{(- 10)}^{2}}{4} - (- 11)} \\ x = 5 \pm \sqrt{\frac{}{4} + 11} \\ x = 5 \pm \sqrt{25 + 11} \\ x = 5 \pm \sqrt{36} \\ x = 5 \pm 6 \\ x_{1} = 5 + 6 \\ x_{1} = 11 \\ x_{2} = 5 - 6 \\ x_{2} = - 1 \end{matrix}$
$\begin{matrix} (x - 1) (x + 2) - (2 x - 3) (x + 4) - x + 14 & = 0 \\ x^{2} + x - 2 - (2 x^{2} + 8 x - 3 x - 12) - x + 14 & = 0 \\ x^{2} - 2 x^{2} - 5 x + 12 + 12 & = 0 \\ - x^{2} - 5 x + 24 & = 0 \\ x^{2} + 5 x - 24 & = 0 \\ m = 5 \\ n = - 24 \\ x = - \frac{5}{2} \pm \sqrt{\frac{5^{2}}{4} - (- 24)} \\ x = - \frac{5}{2} \pm \sqrt{\frac{25}{4} + 24} \\ x = - \frac{5}{2} \pm \sqrt{\frac{25 + 96}{4}} \\ x = - \frac{5}{2} \pm \sqrt{\frac{121}{4}} \\ x = - \frac{5}{2} \pm \frac{11}{2} \\ x_{1} = - \frac{5}{2} + \frac{11}{2} = \frac{- 5 + 11}{2} = \frac{}{2} \\ x_{1} = 3 \\ x_{2} = - \frac{5}{2} - \frac{11}{2} = \frac{- 5 - 11}{2} = - \frac{}{2} \\ x_{2} = - 8 \end{matrix}$