Ejercicio 268 – Solucionario Baldor

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DESCARGA

CAPITULO XXXIII

Ecuaciones de segundo grado con denominadores

Ejercicio 268

Resolver las siguientes ecuaciones:

$\begin{matrix} \frac{x^{2}}{5} - \frac{x}{2} & = \frac{3}{10} \\ \frac{x^{2}}{5} - \frac{x}{2} & = \frac{3}{10} \\ \frac{2 x^{2} - 5 x}{10} & = \frac{3}{10} \\ 2 x^{2} - 5 x - 3 & = 0 \\ x = \frac{5 \pm \sqrt{{(- 5)}^{2} - 4 (2) (- 3)}}{2 (2)} \\ x = \frac{5 \pm \sqrt{25 + 24}}{4} \\ x = \frac{5 \pm \sqrt{49}}{4} \\ x = \frac{5 \pm 7}{4} \\ x_{1} = \frac{5 + 7}{4} \\ x_{1} = \frac{}{4} \\ x_{1} = 3 \\ x_{2} = \frac{5 - 7}{4} \\ x_{2} = - \frac{2}{} \\ x_{2} = \frac{1}{2} \end{matrix}$
$\begin{matrix} 4 x - \frac{13}{x} & = \frac{3}{2} \\ 4 x - \frac{13}{x} - \frac{3}{2} & = 0 \\ \frac{8 x^{2} - 26 - 3 x}{2 x} & = 0 \\ 8 x^{2} - 3 x - 26 & = 0 \\ x = \frac{3 \pm \sqrt{{(- 3)}^{2} - 4 (8) (- 26)}}{2 (8)} \\ x = \frac{3 \pm \sqrt{9 + 832}}{16} \\ x = \frac{3 \pm \sqrt{841}}{16} \\ x = \frac{3 \pm 29}{16} \\ x_{1} = \frac{3 + 29}{16} \\ x_{1} = \frac{}{16} \\ x_{1} = 2 \\ x_{2} = \frac{3 - 29}{16} \\ x_{2} = - \frac{}{} \\ x_{2} = - \frac{13}{8} \end{matrix}$
$\begin{matrix} \frac{x^{2}}{6} - \frac{x}{2} & = 3 (x - 5) \\ \frac{x^{2} - 3 x}{6} & = 3 (x - 5) \\ x^{2} - 3 x & = 18 (x - 5) \\ x^{2} - 3 x & = 18 x - 90 \\ x^{2} - 3 x - 18 x + 90 & = 0 \\ x^{2} - 21 x + 90 & = 0 \\ m = - 21 \\ n = 90 \\ x = \frac{21}{2} \pm \sqrt{\frac{{(- 21)}^{2}}{4} - 90} \\ x = \frac{21}{2} \pm \sqrt{\frac{441}{4} - 90} \\ x = \frac{21}{2} \pm \sqrt{\frac{441 - 360}{4}} \\ x = \frac{21}{2} \pm \sqrt{\frac{81}{4}} \\ x = \frac{21}{2} \pm \frac{9}{2} \\ x_{1} = \frac{21}{2} + \frac{9}{2} = \frac{}{2} \\ x_{1} = 15 \\ x_{2} = \frac{21}{2} - \frac{9}{2} = \frac{}{2} \\ x_{2} = 6 \end{matrix}$
$\begin{matrix} \frac{1}{4} (x - 4) + \frac{2}{5} (x - 5) & = \frac{1}{5} (x^{2} - 53) \\ M u l t i p l i c o l a e c u a c i ó n p o r 20 \\ 5 (x - 4) + 8 (x - 5) & = 4 (x^{2} - 53) \\ 5 x - 20 + 8 x - 40 & = 4 x^{2} - 212 \\ 0 & = 4 x^{2} - 212 - 13 x + 60 \\ 4 x^{2} - 13 x - 152 & = 0 \\ x = \frac{13 \pm \sqrt{{(- 13)}^{2} - 4 (4) (- 152)}}{2 (4)} \\ x = \frac{13 \pm \sqrt{169 + 2432}}{8} \\ x = \frac{13 \pm \sqrt{2601}}{8} \\ x = \frac{13 \pm 51}{8} \\ x_{1} = \frac{13 + 51}{8} \\ x_{1} = \frac{}{8} \\ x_{1} = 8 \\ x_{2} = \frac{13 - 51}{8} \\ x_{2} = - \frac{}{} \\ x_{2} = - \frac{19}{4} \end{matrix}$
$\begin{matrix} \frac{5}{x} - \frac{1}{x + 2} & = 1 \\ \frac{5 (x + 2) - x}{x (x + 2)} & = 1 \\ 5 (x + 2) - x & = x (x + 2) \\ 5 x + 10 - x & = x^{2} + 2 x \\ 0 & = x^{2} + 2 x - 4 x - 10 \\ x^{2} - 2 x - 10 & = 0 \\ m = - 2 \\ n = - 10 \\ x = \frac{2}{2} \pm \sqrt{\frac{2^{2}}{4} - (- 10)} \\ x = 1 \pm \sqrt{1 + 10} \\ x = 1 \pm \sqrt{11} \\ x_{1} = 1 + \sqrt{11} \\ x_{2} = 1 - \sqrt{11} \end{matrix}$
$\begin{matrix} \frac{15}{x} - \frac{11 x + 5}{x^{2}} & = - 1 \\ \frac{15 x - 11 x - 5}{x^{2}} & = - 1 \\ 4 x - 5 & = - x^{2} \\ x^{2} + 4 x - 5 & = 0 \\ m = 4 \\ n = - 5 \\ x = - \frac{}{2} \pm \sqrt{\frac{4^{2}}{4} - (- 5)} \\ x = - 2 \pm \sqrt{4 + 5} \\ x = - 2 \pm \sqrt{9} \\ x = - 2 \pm 3 \\ x_{1} = - 2 + 3 \\ x_{1} = 1 \\ x_{2} = - 2 - 3 \\ x_{2} = - 5 \end{matrix}$
$\begin{matrix} \frac{8 x}{3 x + 5} + \frac{5 x - 1}{x + 1} & = 3 \\ \frac{8 x (x + 1) + (5 x - 1) (3 x + 5)}{(3 x + 5) (x + 1)} & = 3 \\ 8 x (x + 1) + (5 x - 1) (3 x + 5) & = 3 (3 x + 5) (x + 1) \\ 8 x^{2} + 8 x + 15 x^{2} + 25 x - 3 x - 5 & = 3 (3 x^{2} + 3 x + 5 x + 5) \\ 23 x^{2} + 30 x - 5 & = 9 x^{2} + 24 x + 15 \\ 23 x^{2} + 30 x - 5 - 9 x^{2} - 24 x - 15 & = 0 \\ 14 x^{2} + 6 x - 20 & = 0 \\ D i v i d i e n d o l a e c u a c i ó n p a r a 2 \\ 7 x^{2} + 3 x - 10 & = 0 \\ x = \frac{- 3 \pm \sqrt{3^{3} - 4 (7) (- 10)}}{2 (7)} \\ x = \frac{- 3 \pm \sqrt{9 + 280}}{14} \\ x = \frac{- 3 \pm \sqrt{289}}{14} \\ x = \frac{- 3 \pm 17}{14} \\ x_{1} = \frac{- 3 + 17}{14} = \frac{14}{14} \\ x_{1} = 1 \\ x_{2} = \frac{- 3 - 17}{14} = - \frac{}{} \\ x_{2} = - \frac{10}{7} \end{matrix}$
$\begin{matrix} \frac{1}{x - 2} - \frac{1}{x - 1} & = \frac{1}{6} \\ \frac{x - 1 - x + 2}{(x - 2) (x - 1)} & = \frac{1}{6} \\ \frac{1}{(x - 2) (x - 1)} & = \frac{1}{6} \\ 6 & = (x - 2) (x - 1) \\ 6 & = x^{2} - 3 x + 2 \\ 0 & = x^{2} - 3 x + 2 - 6 \\ x^{2} - 3 x - 4 & = 0 \\ m = - 3 \\ n = - 4 \\ x = \frac{3}{2} \pm \sqrt{\frac{{(- 3)}^{2}}{4} - (- 4)} \\ x = \frac{3}{2} \pm \sqrt{\frac{9}{4} + 4} \\ x = \frac{3}{2} \pm \sqrt{\frac{9 + 16}{4}} \\ x = \frac{3}{2} \pm \sqrt{\frac{25}{4}} \\ x = \frac{3}{2} \pm \frac{5}{2} \\ x_{1} = \frac{3}{2} + \frac{5}{2} = \frac{3 + 5}{2} = \frac{}{2} \\ x_{1} = 4 \\ x_{2} = \frac{3}{2} - \frac{5}{2} = \frac{3 - 5}{2} = - \frac{2}{2} \\ x_{2} = - 1 \end{matrix}$
$\begin{matrix} 1 - \frac{2 x - 3}{x + 5} & = \frac{x - 2}{10} \\ \frac{x + 5 - 2 x + 3}{x + 5} & = \frac{x - 2}{10} \\ \frac{8 - x}{x + 5} & = \frac{x - 2}{10} \\ 10 (8 - x) & = (x - 2) (x + 5) \\ 80 - 10 x & = x^{2} + 3 x - 10 \\ 0 & = x^{2} + 3 x - 10 - 80 + 10 x \\ x^{2} + 13 x - 90 & = 0 \\ m = 13 \\ n = - 90 \\ x = - \frac{13}{2} \pm \sqrt{\frac{{(- 13)}^{2}}{4} - (- 90)} \\ x = - \frac{13}{2} \pm \sqrt{\frac{169}{4} + 90} \\ x = - \frac{13}{2} \pm \sqrt{\frac{169 + 360}{4}} \\ x = - \frac{13}{2} \pm \sqrt{\frac{529}{4}} \\ x = - \frac{13}{2} \pm \frac{23}{2} \\ x_{1} = - \frac{13}{2} + \frac{23}{2} = \frac{- 13 + 23}{2} = \frac{}{2} \\ x_{1} = 5 \\ x_{2} = - \frac{13}{2} - \frac{23}{2} = \frac{- 13 - 23}{2} = - \frac{}{2} \\ x_{2} = - 18 \end{matrix}$
$\begin{matrix} \frac{x - 13}{x} & = 5 - \frac{10 (5 x + 3)}{x^{2}} \\ \frac{x - 13}{x} & = \frac{5 x^{2} - 10 (5 x + 3)}{x^{2}} \\ x (x - 13) & = 5 x^{2} - 50 x - 30 \\ x^{2} - 13 x & = 5 x^{2} - 50 x - 30 \\ 0 & = 5 x^{2} - 50 x - 30 - x^{2} + 13 x \\ 4 x^{2} - 37 x - 30 & = 0 \\ x = \frac{37 \pm \sqrt{{(- 37)}^{2} - 4 (4) (- 30)}}{2 (4)} \\ x = \frac{37 \pm \sqrt{1369 + 480}}{8} \\ x = \frac{37 \pm \sqrt{1849}}{8} \\ x = \frac{37 \pm 43}{8} \\ x_{1} = \frac{37 + 43}{8} = \frac{}{8} \\ x_{1} = 10 \\ x_{2} = \frac{37 - 43}{8} = - \frac{}{} \\ x_{2} = - \frac{3}{4} \end{matrix}$
$\begin{matrix} \frac{x}{x - 2} - \frac{x - 2}{x} & = \frac{5}{2} \\ \frac{x^{2} - {(x - 2)}^{2}}{x (x - 2)} & = \frac{5}{2} \\ \frac{x^{2} - (x^{2} - 4 x + 4)}{x^{2} - 2 x} & = \frac{5}{2} \\ \frac{x^{2} - x^{2} + 4 x - 4}{x^{2} - 2 x} & = \frac{5}{2} \\ 2 (4 x - 4) & = 5 (x^{2} - 2 x) \\ 8 x - 8 & = 5 x^{2} - 10 x \\ 0 & = 5 x^{2} - 10 x - 8 x + 8 \\ 5 x^{2} - 18 x + 8 & = 0 \\ x = \frac{18 \pm \sqrt{{(- 18)}^{2} - 4 (5) (8)}}{2 (5)} \\ x = \frac{18 \pm \sqrt{324 - 160}}{10} \\ x = \frac{18 \pm \sqrt{164}}{10} \\ x = \frac{18 \pm \sqrt{2^{2} .41}}{10} \\ x = \frac{18 \pm 2 \sqrt{41}}{10} \\ x = \frac{2 (9 \pm \sqrt{41})}{} \\ x_{1} = \frac{9 + \sqrt{41}}{5} \\ x_{2} = \frac{9 - \sqrt{41}}{5} \end{matrix}$
$\begin{matrix} \frac{4 x^{2}}{x - 1} - \frac{1 - 3 x}{4} & = \frac{20 x}{3} \\ \frac{16 x^{2} - (x - 1) (1 - 3 x)}{4 (x - 1)} & = \frac{20 x}{3} \\ \frac{16 x^{2} - (x - 3 x^{2} - 1 + 3 x)}{4 x - 4} & = \frac{20 x}{3} \\ \frac{16 x^{2} + 3 x^{2} - 4 x + 1}{4 x - 4} & = \frac{20 x}{3} \\ 3 (19 x^{2} - 4 x + 1) & = 20 x (4 x - 4) \\ 57 x^{2} - 12 x + 3 & = 80 x^{2} - 80 x \\ 0 & = 80 x^{2} - 80 x - 57 x^{2} + 12 x - 3 \\ 23 x^{2} - 68 x - 3 & = 0 \\ x = \frac{68 \pm \sqrt{{(- 68)}^{2} - 4 (23) (- 3)}}{2 (23)} \\ x = \frac{68 \pm \sqrt{4624 + 276}}{46} \\ x = \frac{68 \pm \sqrt{4900}}{46} \\ x = \frac{68 \pm 70}{46} \\ x_{1} = \frac{68 + 70}{46} = \frac{}{46} \\ x_{1} = 3 \\ x_{2} = \frac{68 - 70}{46} = - \frac{2}{} \\ x_{2} = - \frac{1}{23} \end{matrix}$
$\begin{matrix} \frac{3 x - 1}{x} - \frac{2 x}{2 x - 1} - \frac{7}{6} & = 0 \\ \frac{6 (2 x - 1) (3 x - 1) - 12 x^{2} - 7 x (2 x - 1)}{6 x (2 x - 1)} & = 0 \\ 6 (2 x - 1) (3 x - 1) - 12 x^{2} - 7 x (2 x - 1) & = 0 \\ 6 (6 x^{2} - 2 x - 3 x + 1) - 12 x^{2} - 14 x^{2} + 7 x & = 0 \\ 36 x^{2} - 30 x + 6 - 26 x^{2} + 7 x & = 0 \\ 10 x^{2} - 23 x + 6 & = 0 \\ x = \frac{23 \pm \sqrt{{(- 23)}^{2} - 4 (10) (6)}}{2 (10)} \\ x = \frac{23 \pm \sqrt{529 - 240}}{20} \\ x = \frac{23 \pm \sqrt{289}}{20} \\ x = \frac{23 \pm 17}{20} \\ x_{1} = \frac{23 + 17}{20} = \frac{}{20} \\ x_{1} = 2 \\ x_{2} = \frac{23 - 17}{20} = \frac{}{} \\ x_{2} = \frac{3}{10} \end{matrix}$
$\begin{matrix} \frac{5 x - 8}{x - 1} & = \frac{7 x - 4}{x + 2} \\ (5 x - 8) (x + 2) & = (7 x - 4) (x - 1) \\ 5 x^{2} + 10 x - 8 x - 16 & = 7 x^{2} - 7 x - 4 x + 4 \\ 0 & = 7 x^{2} - 11 x + 4 - 5 x^{2} - 2 x + 16 \\ 2 x^{2} - 13 x + 20 & = 0 \\ x = \frac{13 \pm \sqrt{{(- 13)}^{2} - 4 (2) (20)}}{2 (2)} \\ x = \frac{13 \pm \sqrt{169 - 160}}{4} \\ x = \frac{13 \pm \sqrt{9}}{4} \\ x = \frac{13 \pm 3}{4} \\ x_{1} = \frac{13 + 3}{4} = \frac{}{4} \\ x_{1} = 4 \\ x_{2} = \frac{13 - 3}{4} = \frac{}{} \\ x_{2} = \frac{5}{2} \end{matrix}$
$\begin{matrix} \frac{x + 3}{2 x - 1} - \frac{5 x - 1}{4 x + 7} & = 0 \\ \frac{(x + 3) (4 x + 7) - (5 x - 1) (2 x - 1)}{(2 x - 1) (4 x + 7)} & = 0 \\ (x + 3) (4 x + 7) - (5 x - 1) (2 x - 1) & = 0 \\ 4 x^{2} + 7 x + 12 x + 21 - (10 x^{2} - 5 x - 2 x + 1) & = 0 \\ 4 x^{2} + 19 x + 21 - 10 x^{2} + 7 x - 1 & = 0 \\ - 6 x^{2} + 26 x + 20 & = 0 \\ D i v i d i e n d o l a e c u a c i ó n p a r a - 2 \\ 3 x^{2} - 13 x - 10 & = 0 \\ x = \frac{13 \pm \sqrt{{(- 13)}^{2} - 4 (3) (- 10)}}{2 (3)} \\ x = \frac{13 \pm \sqrt{169 + 120}}{6} \\ x = \frac{13 \pm \sqrt{289}}{6} \\ x = \frac{13 \pm 17}{6} \\ x_{1} = \frac{13 + 17}{6} = \frac{}{6} \\ x_{1} = 5 \\ x_{2} = \frac{13 - 17}{6} = - \frac{}{} \\ x_{2} = - \frac{2}{3} \end{matrix}$
$\begin{matrix} \frac{1}{4 - x} - \frac{1}{6} & = \frac{1}{x + 1} \\ \frac{6 - 4 + x}{6 (4 - x)} & = \frac{1}{x + 1} \\ \frac{2 + x}{24 - 6 x} & = \frac{1}{x + 1} \\ (2 + x) (x + 1) & = 24 - 6 x \\ x^{2} + 3 x + 2 & = 24 - 6 x \\ x^{2} + 3 x + 2 - 24 + 6 x & = 0 \\ x^{2} + 9 x - 22 & = 0 \\ m = 9 \\ n = - 22 \\ x = - \frac{9}{2} \pm \sqrt{\frac{9^{2}}{4} - (- 22)} \\ x = - \frac{9}{2} \pm \sqrt{\frac{81}{4} + 22} \\ x = - \frac{9}{2} \pm \sqrt{\frac{81 + 88}{4}} \\ x = - \frac{9}{2} \pm \sqrt{\frac{169}{4}} \\ x = - \frac{9}{2} \pm \frac{13}{2} \\ x_{1} = - \frac{9}{2} + \frac{13}{2} = \frac{}{2} \\ x_{1} = 2 \\ x_{2} = - \frac{9}{2} - \frac{13}{2} = - \frac{}{2} \\ x_{2} = - 11 \end{matrix}$
$\begin{matrix} \frac{x + 4}{x + 5} - \frac{x + 2}{x + 3} & = \frac{1}{24} \\ \frac{(x + 4) (x + 3) - (x + 2) (x + 5)}{(x + 5) (x + 3)} & = \frac{1}{24} \\ \frac{x^{2} + 7 x + 12 - (x^{2} + 7 x + 10)}{x^{2} + 8 x + 15} & = \frac{1}{24} \\ \frac{x^{2} + 7 x + 12 - x^{2} - 7 x - 10}{x^{2} + 8 x + 15} & = \frac{1}{24} \\ \frac{2}{x^{2} + 8 x + 15} & = \frac{1}{24} \\ 48 & = x^{2} + 8 x + 15 \\ 0 & = x^{2} + 8 x + 15 - 48 \\ x^{2} + 8 x - 33 & = 0 \\ m = 8 \\ n = - 33 \\ x = - \frac{}{2} \pm \sqrt{\frac{8^{2}}{4} - (- 33)} \\ x = - 4 \pm \sqrt{\frac{}{4} + 33} \\ x = - 4 \pm \sqrt{16 + 33} \\ x = - 4 \pm \sqrt{49} \\ x = - 4 \pm 7 \\ x_{1} = - 4 + 7 \\ x_{1} = 3 \\ x_{2} = - 4 - 7 \\ x_{2} = - 11 \end{matrix}$
$\begin{matrix} \frac{5}{x^{2} - 1} - \frac{6}{x + 1} & = 3 \frac{5}{8} \\ \frac{5}{(x + 1) (x - 1)} - \frac{6}{x + 1} & = \frac{29}{8} \\ \frac{5 - 6 (x - 1)}{(x + 1) (x - 1)} & = \frac{29}{8} \\ \frac{5 - 6 x + 6}{x^{2} - 1} & = \frac{29}{8} \\ \frac{11 - 6 x}{x^{2} - 1} & = \frac{29}{8} \\ 8 (11 - 6 x) & = 29 (x^{2} - 1) \\ 88 - 48 x & = 29 x^{2} - 29 \\ 0 & = 29 x^{2} - 29 - 88 + 48 x \\ 29 x^{2} + 48 x - 117 & = 0 \\ x = \frac{- 48 \pm \sqrt{4 8^{2} - 4 (29) (- 117)}}{2 (29)} \\ x = \frac{- 48 \pm \sqrt{2304 + 13572}}{58} \\ x = \frac{- 48 \pm \sqrt{15876}}{58} \\ x = \frac{- 48 \pm 126}{58} \\ x_{1} = \frac{- 48 + 126}{58} \\ x_{1} = \frac{}{} \\ x_{1} = \frac{39}{29} \\ x_{2} = \frac{- 48 - 126}{58} \\ x_{2} = \frac{-}{58} \\ x_{2} = - 3 \end{matrix}$
$\begin{matrix} \frac{x - 1}{x + 1} + \frac{x + 1}{x - 1} & = \frac{2 x + 9}{x + 3} \\ \frac{{(x - 1)}^{2} + {(x + 1)}^{2}}{(x + 1) (x - 1)} & = \frac{2 x + 9}{x + 3} \\ \frac{{(x - 1)}^{2} + {(x + 1)}^{2}}{x^{2} - 1} & = \frac{2 x + 9}{x + 3} \\ \frac{x^{2} - 2 x + 1 + x^{2} + 2 x + 1}{x^{2} - 1} & = \frac{2 x + 9}{x + 3} \\ \frac{2 x^{2} + 2}{x^{2} - 1} & = \frac{2 x + 9}{x + 3} \\ (2 x^{2} + 2) (x + 3) & = (x^{2} - 1) (2 x + 9) \\ 2 x^{3} + 2 x + 6 x^{2} + 6 & = 2 x^{3} + 9 x^{2} - 2 x - 9 \\ 0 & = 9 x^{2} - 2 x - 9 - 2 x - 6 x^{2} - 6 \\ 3 x^{2} - 4 x - 15 & = 0 \\ x = \frac{4 \pm \sqrt{{(- 4)}^{2} - 4 (3) (- 15)}}{2 (3)} \\ x = \frac{4 \pm \sqrt{16 + 180}}{6} \\ x = \frac{4 \pm \sqrt{196}}{6} \\ x = \frac{4 \pm 14}{6} \\ x_{1} = \frac{4 + 14}{6} \\ x_{1} = \frac{}{6} \\ x_{1} = 3 \\ x_{2} = \frac{4 - 14}{6} \\ x_{2} = \frac{-}{} \\ x_{2} = - \frac{5}{3} \end{matrix}$
$\begin{matrix} \frac{3}{x + 2} - \frac{1}{x - 2} & = \frac{1}{x + 1} \\ \frac{3 (x - 2) - (x + 2)}{(x + 2) (x - 2)} & = \frac{1}{x + 1} \\ \frac{3 x - 6 - x - 2}{x^{2} - 4} & = \frac{1}{x + 1} \\ \frac{2 x - 8}{x^{2} - 4} & = \frac{1}{x + 1} \\ (2 x - 8) (x + 1) & = x^{2} - 4 \\ 2 x^{2} - 8 x + 2 x - 8 & = x^{2} - 4 \\ 2 x^{2} - 6 x - 8 - x^{2} + 4 & = 0 \\ x^{2} - 6 x - 4 & = 0 \\ m = - 6 \\ n = - 4 \\ x = \frac{}{2} \pm \sqrt{\frac{{(- 6)}^{2}}{4} - (- 4)} \\ x = 3 \pm \sqrt{\frac{}{4} + 4} \\ x = 3 \pm \sqrt{9 + 4} \\ x = 3 \pm \sqrt{13} \\ x_{1} = 3 + \sqrt{13} \\ x_{2} = 3 - \sqrt{13} \end{matrix}$