▶️ Ejercicio 269 álgebra - Solucionario Baldor ✅✅

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CAPITULO XXXIII

Ecuaciones de segundo grado por descomposición en factores

Ejercicio 269

Resolver por descomposición en factores:

$\begin{matrix} x^{2} - x - 6 & = 0 \\ (x - 3) (x + 2) & = 0 \\ x - 3 & = 0 \\ x_{1} & = 3 \\ x + 2 & = 0 \\ x_{2} & = - 2 \end{matrix}$
$\begin{matrix} x^{2} + 7 x & = 18 \\ x^{2} + 7 x - 18 & = 0 \\ (x + 9) (x - 2) & = 0 \\ x + 9 & = 0 \\ x_{1} & = - 9 \\ x - 2 & = 0 \\ x_{2} & = 2 \end{matrix}$
$\begin{matrix} 8 x - 65 & = - x^{2} \\ x^{2} + 8 x - 65 & = 0 \\ (x + 13) (x - 5) & = 0 \\ x + 13 & = 0 \\ x_{1} & = - 13 \\ x - 5 & = 0 \\ x_{2} & = 5 \end{matrix}$
$\begin{matrix} x^{2} & = 108 - 3 x \\ x^{2} + 3 x - 108 & = 0 \\ (x + 12) (x - 9) & = 0 \\ x + 12 & = 0 \\ x_{1} & = - 12 \\ x - 9 & = 0 \\ x_{2} & = 9 \end{matrix}$
$\begin{matrix} 2 x^{2} + 7 x - 4 & = 0 \\ 2 x^{2} - x + 8 x - 4 & = 0 \\ x (2 x - 1) + 4 (2 x - 1) & = 0 \\ (2 x - 1) (x + 4) & = 0 \\ x + 4 & = 0 \\ x_{1} & = - 4 \\ 2 x - 1 & = 0 \\ 2 x & = 1 \\ x_{2} & = \frac{1}{2} \end{matrix}$
$\begin{matrix} 6 x^{2} & = 10 - 11 x \\ 6 x^{2} + 11 x - 10 & = 0 \\ 6 x^{2} - 4 x + 15 x - 10 & = 0 \\ 2 x (3 x - 2) + 5 (3 x - 2) & = 0 \\ (3 x - 2) (2 x + 5) & = 0 \\ 3 x - 2 & = 0 \\ x_{1} & = \frac{2}{3} \\ 2 x + 5 & = 0 \\ x_{2} & = - \frac{5}{2} \end{matrix}$
$\begin{matrix} 20 x^{2} - 27 x & = 14 \\ 20 x^{2} - 27 x - 14 & = 0 \\ 20 x^{2} - 35 x + 8 x - 14 & = 0 \\ 5 x (4 x - 7) + 2 (4 x - 7) & = 0 \\ (4 x - 7) (5 x + 2) & = 0 \\ 4 x - 7 & = 0 \\ x_{1} & = \frac{7}{4} \\ 5 x + 2 & = 0 \\ x_{2} & = - \frac{2}{5} \end{matrix}$
$\begin{matrix} 7 x & = 15 - 30 x^{2} \\ 30 x^{2} + 7 x - 15 & = 0 \\ 30 x^{2} - 18 x + 25 x - 15 & = 0 \\ 6 x (5 x - 3) + 5 (5 x - 3) & = 0 \\ (5 x - 3) (6 x + 5) & = 0 \\ 6 x + 5 & = 0 \\ x_{1} & = - \frac{5}{6} \\ 5 x - 3 & = 0 \\ x_{2} & = \frac{3}{5} \end{matrix}$
$\begin{matrix} 60 & = 8 x^{2} + 157 x \\ 8 x^{2} + 157 x - 60 & = 0 \\ 8 x^{2} + 160 x - 3 x - 60 & = 0 \\ 8 x (x + 20) - 3 (x + 20) & = 0 \\ (x + 20) (8 x - 3) & = 0 \\ x + 20 & = 0 \\ x_{1} & = - 20 \\ 8 x - 3 & = 0 \\ x_{2} & = \frac{3}{8} \end{matrix}$
$\begin{matrix} x (x - 1) - 5 (x - 2) & = 2 \\ x^{2} - x - 5 x + 10 - 2 & = 0 \\ x^{2} - 6 x + 8 & = 0 \\ (x - 4) (x - 2) & = 0 \\ x - 4 & = 0 \\ x_{1} & = 4 \\ x - 2 & = 0 \\ x_{2} & = 2 \end{matrix}$
$\begin{matrix} {(x - 2)}^{2} - {(2 x + 3)}^{2} & = - 80 \\ [(x - 2) + (2 x + 3)] [(x - 2) - (2 x + 3)] & = - 80 \\ [x - 2 + 2 x + 3] [x - 2 - 2 x - 3] & = - 80 \\ (3 x + 1) (- x - 5) & = - 80 \\ - 3 x^{2} - 15 x - x - 5 + 80 & = 0 \\ - 3 x^{2} - 16 x + 75 & = 0 \\ 3 x^{2} + 16 x - 75 & = 0 \\ 3 x^{2} - 9 x + 25 x - 75 & = 0 \\ 3 x (x - 3) + 25 (x - 3) & = 0 \\ (x - 3) (3 x + 25) & = 0 \\ x - 3 & = 0 \\ x_{1} & = 3 \\ 3 x + 25 & = 0 \\ x_{2} & = - \frac{25}{3} \end{matrix}$
$\begin{matrix} \frac{6}{x^{2}} - \frac{9}{x} & = - \frac{4}{3} \\ \frac{6 - 9 x}{x^{2}} & = - \frac{4}{3} \\ 3 (6 - 9 x) & = - 4 x^{2} \\ 4 x^{2} - 27 x + 18 & = 0 \\ 4 x^{2} - 24 x - 3 x + 18 & = 0 \\ 4 x (x - 6) - 3 (x - 6) & = 0 \\ (x - 6) (4 x - 3) & = 0 \\ x - 6 & = 0 \\ x_{1} & = 6 \\ 4 x - 3 & = 0 \\ x_{2} & = \frac{3}{4} \end{matrix}$
$\begin{matrix} \frac{x + 2}{x} + x & = \frac{74}{x} \\ \frac{x + 2 + x^{2}}{x} & = \frac{74}{x} \\ x + 2 + x^{2} - 74 & = 0 \\ x^{2} + x - 72 & = 0 \\ (x + 9) (x - 8) & = 0 \\ x - 8 & = 0 \\ x_{1} & = 8 \\ x + 9 & = 0 \\ x_{2} & = - 9 \end{matrix}$
$\begin{matrix} {(x + 2)}^{2} - \frac{2 x - 5}{3} & = 3 \\ \frac{3 {(x + 2)}^{2} - 2 x + 5}{3} & = 3 \\ 3 (x^{2} + 4 x + 4) - 2 x + 5 & = 9 \\ 3 x^{2} + 12 x + 12 - 2 x + 5 - 9 & = 0 \\ 3 x^{2} + 10 x + 8 & = 0 \\ 3 x^{2} + 6 x + 4 x + 8 & = 0 \\ 3 x (x + 2) + 4 (x + 2) & = 0 \\ (x + 2) (3 x + 4) & = 0 \\ x + 2 & = 0 \\ x_{1} & = - 2 \\ 3 x + 4 & = 0 \\ x_{2} & = - \frac{4}{3} \end{matrix}$
$\begin{matrix} \frac{x}{x - 2} + x & = \frac{3 x + 15}{4} \\ \frac{x + x (x - 2)}{x - 2} & = \frac{3 x + 15}{4} \\ \frac{x + x^{2} - 2 x}{x - 2} & = \frac{3 x + 15}{4} \\ \frac{x^{2} - x}{x - 2} & = \frac{3 x + 15}{4} \\ 4 (x^{2} - x) & = (x - 2) (3 x + 15) \\ 4 x^{2} - 4 x & = 3 x^{2} + 15 x - 6 x - 30 \\ 4 x^{2} - 4 x - 3 x^{2} - 9 x + 30 & = 0 \\ x^{2} - 13 x + 30 & = 0 \\ (x - 10) (x - 3) & = 0 \\ x - 10 & = 0 \\ x_{1} & = 10 \\ x - 3 & = 0 \\ x_{2} & = 3 \end{matrix}$
$\begin{matrix} \frac{6}{x - 4} - \frac{4}{x} & = \frac{5}{12} \\ \frac{6 x - 4 (x - 4)}{x (x - 4)} & = \frac{5}{12} \\ \frac{6 x - 4 x + 16}{x^{2} - 4 x} & = \frac{5}{12} \\ 12 (2 x + 16) & = 5 (x^{2} - 4 x) \\ 24 x + 192 & = 5 x^{2} - 20 x \\ 0 & = 5 x^{2} - 20 x - 24 x - 192 \\ 5 x^{2} - 44 x - 192 & = 0 \\ 5 x^{2} - 60 x + 16 x - 192 & = 0 \\ 5 x (x - 12) + 16 (x - 12) & = 0 \\ (x - 12) (5 x + 16) & = 0 \\ x - 12 & = 0 \\ x_{1} & = 12 \\ 5 x + 16 & = 0 \\ x_{2} & = - \frac{16}{5} \end{matrix}$
$\begin{matrix} {(x - 2)}^{3} - {(x - 3)}^{3} & = 37 \\ [(x - 2) - (x - 3)] [{(x - 2)}^{2} + (x - 2) (x - 3) + {(x - 3)}^{2}] & = 37 \\ [x^{2} - 4 x + 4 + x^{2} - 5 x + 6 + x^{2} - 6 x + 9] & = 37 \\ 3 x^{2} - 15 x + 19 - 37 & = 0 \\ 3 x^{2} - 15 x - 18 & = 0 \\ D i v i d i e n d o l a e c u a c i ó n p a r a 3 \\ x^{2} - 5 x - 6 & = 0 \\ (x - 6) (x + 1) & = 0 \\ x - 6 & = 0 \\ x_{1} & = 6 \\ x + 1 & = 0 \\ x_{2} & = - 1 \end{matrix}$
$\begin{matrix} \frac{x - 1}{x + 1} - 2 & = \frac{x + 3}{3} \\ \frac{x - 1 - 2 (x + 1)}{x + 1} & = \frac{x + 3}{3} \\ \frac{x - 1 - 2 x - 2}{x + 1} & = \frac{x + 3}{3} \\ \frac{- x - 3}{x + 1} & = \frac{x + 3}{3} \\ 3 (- x - 3) & = (x + 3) (x + 1) \\ - 3 x - 9 & = x^{2} + 4 x + 3 \\ - 3 x - 9 - x^{2} - 4 x - 3 & = 0 \\ - x^{2} - 7 x - 12 & = 0 \\ x^{2} + 7 x + 12 & = 0 \\ (x + 4) (x + 3) & = 0 \\ x + 4 & = 0 \\ x_{1} & = - 4 \\ x + 3 & = 0 \\ x_{2} & = - 3 \end{matrix}$
$\begin{matrix} \frac{4 x - 1}{2 x + 3} & = \frac{2 x + 1}{6 x + 5} \\ (4 x - 1) (6 x + 5) & = (2 x + 1) (2 x + 3) \\ 24 x^{2} + 20 x - 6 x - 5 & = 4 x^{2} + 8 x + 3 \\ 24 x^{2} + 14 x - 5 - 4 x^{2} - 8 x - 3 & = 0 \\ 20 x^{2} + 6 x - 8 & = 0 \\ D i v i d i e n d o l a e c u a c i ó n p a r a 2 \\ 10 x^{2} + 3 x - 4 & = 0 \\ 10 x^{2} - 5 x + 8 x - 4 & = 0 \\ 5 x (2 x - 1) + 4 (2 x - 1) & = 0 \\ (2 x - 1) (5 x + 4) & = 0 \\ 2 x - 1 & = 0 \\ x_{1} & = \frac{1}{2} \\ 5 x + 4 & = 0 \\ x_{2} & = - \frac{4}{5} \end{matrix}$
$\begin{matrix} \frac{3 x + 2}{4} & = 5 - \frac{9 x + 14}{12 x} \\ \frac{3 x + 2}{4} & = \frac{60 x - 9 x - 14}{x} \\ 3 x + 2 & = \frac{51 x - 14}{3 x} \\ 3 x (3 x + 2) & = 51 x - 14 \\ 9 x^{2} + 6 x - 51 x + 14 & = 0 \\ 9 x^{2} - 45 x + 14 & = 0 \\ 9 x^{2} - 3 x - 42 x + 14 & = 0 \\ 3 x (3 x - 1) - 14 (3 x - 1) & = 0 \\ (3 x - 1) (3 x - 14) & = 0 \\ 3 x - 1 & = 0 \\ x_{1} & = \frac{1}{3} \\ 3 x - 14 & = 0 \\ x_{2} & = \frac{14}{3} \end{matrix}$

Categories: Capítulo XXXIII