Ejercicio 270

octubre 30, 2019abril 15, 2019 por Gestor Baldor

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CAPITULO XXXIII

Ecuaciones literales de segundo grado

Ejercicio 270

Resolver las ecuaciones:

$\begin{matrix} x^{2} + 2 a x - 35 a^{2} & = 0 \\ (x + 7 a) (x - 5 a) & = 0 \\ x + 7 a & = 0 \\ x_{1} & = - 7 a \\ x - 5 a & = 0 \\ x_{2} & = 5 a \end{matrix}$
$\begin{matrix} 10 x^{2} & = 36 a^{2} - 37 a x \\ 10 x^{2} + 37 a x - 36 a^{2} & = 0 \\ 10 x^{2} + 45 a x - 8 a x - 36 a^{2} & = 0 \\ 5 x (2 x + 9 a) - 4 a (2 x + 9 a) & = 0 \\ (2 x + 9 a) (5 x - 4 a) & = 0 \\ 2 x + 9 a & = 0 \\ 2 x & = - 9 a \\ x_{1} & = - \frac{9 a}{2} \\ 5 x - 4 a & = 0 \\ 5 x & = 4 a \\ x_{2} & = \frac{4 a}{5} \end{matrix}$
$\begin{matrix} a^{2} x^{2} + a b x - 2 b^{2} & = 0 \\ a^{2} x^{2} - a b x + 2 a b x - 2 b^{2} & = 0 \\ a x (a x - b) + 2 b (a x - b) & = 0 \\ (a x - b) (a x + 2 b) & = 0 \\ a x - b & = 0 \\ a x & = b \\ x_{1} & = \frac{b}{a} \\ a x + 2 b & = 0 \\ a x & = - 2 b \\ x_{2} & = - \frac{2 b}{a} \end{matrix}$
$\begin{matrix} 89 b x & = 42 x^{2} + 22 b^{2} \\ 42 x^{2} - 89 b x + 22 b^{2} & = 0 \\ 42 x^{2} - 12 b x - 77 b x + 22 b^{2} & = 0 \\ 6 x (7 x - 2 b) - 11 b (7 x - 2 b) & = 0 \\ (7 x - 2 b) (6 x - 11 b) & = 0 \\ 7 x - 2 b & = 0 \\ 7 x & = 2 b \\ x_{1} & = \frac{2 b}{7} \\ 6 x - 11 b & = 0 \\ 6 x & = 11 b \\ x_{2} & = \frac{11 b}{6} \end{matrix}$
$\begin{matrix} x^{2} + a x & = 20 a^{2} \\ x^{2} + a x - 20 a^{2} & = 0 \\ (x + 5 a) (x - 4 a) & = 0 \\ x + 5 a & = 0 \\ x_{1} & = - 5 a \\ x - 4 a & = 0 \\ x_{2} & = 4 a \end{matrix}$
$\begin{matrix} 2 x^{2} & = a b x + 3 a^{2} b^{2} \\ 2 x^{2} - a b x - 3 a^{2} b^{2} & = 0 \\ 2 x^{2} + 2 a b x - 3 a b x - 3 a^{2} b^{2} & = 0 \\ 2 x (x + a b) - 3 a b (x + a b) & = 0 \\ (x + a b) (2 x - 3 a b) & = 0 \\ x + a b & = 0 \\ x_{1} & = - a b \\ 2 x - 3 a b & = 0 \\ 2 x & = 3 a b \\ x_{2} & = \frac{3 a b}{2} \end{matrix}$
$\begin{matrix} b^{2} x^{2} + 2 a b x & = 3 a^{2} \\ b^{2} x^{2} + 2 a b x - 3 a^{2} & = 0 \\ b^{2} x^{2} - a b x + 3 a b x - 3 a^{2} & = 0 \\ b x (b x - a) + 3 a (b x - a) & = 0 \\ (b x - a) (b x + 3 a) & = 0 \\ b x - a & = 0 \\ b x & = a \\ x_{1} & = \frac{a}{b} \\ b x + 3 a & = 0 \\ b x & = - 3 a \\ x_{2} & = - \frac{3 a}{b} \end{matrix}$
$\begin{matrix} x^{2} + a x - b x & = a b \\ x^{2} + a x - b x - a b & = 0 \\ x (x + a) - b (x + a) & = 0 \\ (x + a) (x - b) & = 0 \\ x + a & = 0 \\ x_{1} & = - a \\ x - b & = 0 \\ x_{2} & = b \end{matrix}$
$\begin{matrix} x^{2} - 2 a x & = 6 a b - 3 b x \\ x^{2} - 2 a x + 3 b x - 6 a b & = 0 \\ x (x - 2 a) + 3 b (x - 2 a) & = 0 \\ (x - 2 a) (x + 3 b) & = 0 \\ x - 2 a & = 0 \\ x_{1} & = 2 a \\ x + 3 b & = 0 \\ x_{2} & = - 3 b \end{matrix}$
$\begin{matrix} 3 (2 x^{2} - m x) + 4 n x - 2 m n & = 0 \\ 3 x (2 x - m) + 2 n (2 x - m) & = 0 \\ (3 x + 2 n) (2 x - m) & = 0 \\ 3 x + 2 n & = 0 \\ 3 x & = - 2 n \\ x_{1} & = - \frac{2 n}{3} \\ 2 x - m & = 0 \\ 2 x & = m \\ x_{2} & = \frac{m}{2} \end{matrix}$
$\begin{matrix} x^{2} - a^{2} - b x - a b & = 0 \\ (x - a) (x + a) - b (x + a) & = 0 \\ (x + a) (x - a - b) & = 0 \\ x + a & = 0 \\ x_{1} & = - a \\ x - a - b & = 0 \\ x_{2} & = a + b \end{matrix}$
$\begin{matrix} a b x^{2} - x (b - 2 a) & = 2 \\ a b x^{2} - b x + 2 a x - 2 & = 0 \\ b x (a x - 1) + 2 (a x - 1) & = 0 \\ (a x - 1) (b x + 2) & = 0 \\ a x - 1 & = 0 \\ a x & = 1 \\ x_{1} & = \frac{1}{a} \\ b x + 2 & = 0 \\ b x & = - 2 \\ x_{2} & = - \frac{2}{b} \end{matrix}$
$\begin{matrix} x^{2} - 2 a x + a^{2} - b^{2} & = 0 \\ {(x - a)}^{2} - b^{2} & = 0 \\ [(x - a) + b] [(x - a) - b] & = 0 \\ (x - a + b) (x - a - b) & = 0 \\ x - a + b & = 0 \\ x_{1} & = a - b \\ x - a - b & = 0 \\ x_{2} & = a + b \end{matrix}$
$\begin{matrix} 4 x (x - b) + b^{2} & = 4 m^{2} \\ 4 x^{2} - 4 x b + b^{2} - 4 m^{2} & = 0 \\ {(2 x - b)}^{2} - 4 m^{2} & = 0 \\ [(2 x - b) + 2 m] [(2 x - b) - 2 m] & = 0 \\ (2 x - b + 2 m) (2 x - b - 2 m) & = 0 \\ 2 x - b - 2 m & = 0 \\ 2 x & = b + 2 m \\ x_{1} & = \frac{b + 2 m}{2} \\ 2 x - b + 2 m & = 0 \\ 2 x & = b - 2 m \\ x_{2} & = \frac{b - 2 m}{2} \end{matrix}$
$\begin{matrix} x^{2} - b^{2} + 4 a^{2} - 4 a x & = 0 \\ x^{2} - 4 a x + 4 a^{2} - b^{2} & = 0 \\ {(x - 2 a)}^{2} - b^{2} & = 0 \\ [(x - 2 a) + b] [(x - 2 a) - b] & = 0 \\ (x - 2 a + b) (x - 2 a - b) & = 0 \\ x - 2 a - b & = 0 \\ x_{1} & = 2 a + b \\ x - 2 a + b & = 0 \\ x_{2} & = 2 a - b \end{matrix}$
$\begin{matrix} x^{2} - (a + 2) x & = - 2 a \\ x^{2} - a x - 2 x + 2 a & = 0 \\ x (x - a) - 2 (x - a) & = 0 \\ (x - a) (x - 2) & = 0 \\ x - a & = 0 \\ x_{1} & = a \\ x - 2 & = 0 \\ x_{1} & = 2 \end{matrix}$
$\begin{matrix} x^{2} + 2 x (4 - 3 a) & = 48 a \\ x^{2} + 8 x - 6 a x - 48 a & = 0 \\ x (x + 8) - 6 a (x + 8) & = 0 \\ (x + 8) (x - 6 a) & = 0 \\ x + 8 & = 0 \\ x_{1} & = - 8 \\ x - 6 a & = 0 \\ x_{2} & = 6 a \end{matrix}$
$\begin{matrix} x^{2} - 2 x & = m^{2} + 2 m \\ x^{2} - m^{2} - 2 m - 2 x & = 0 \\ (x + m) (x - m) - 2 (m + x) & = 0 \\ (x + m) [(x - m) - 2] & = 0 \\ (x + m) (x - m - 2) & = 0 \\ x + m & = 0 \\ x_{1} & = - m \\ x - m - 2 & = 0 \\ x_{2} & = m + 2 \end{matrix}$
$\begin{matrix} x^{2} + m^{2} x (m - 2) & = 2 m^{5} \\ x^{2} + m^{3} x - 2 m^{2} x - 2 m^{5} & = 0 \\ x (x + m^{3}) - 2 m^{2} (x + m^{3}) & = 0 \\ (x - 2 m^{2}) (x + m^{3}) & = 0 \\ x + m^{3} & = 0 \\ x_{1} & = - m^{3} \\ x - 2 m^{2} & = 0 \\ x_{2} & = 2 m^{2} \end{matrix}$
$\begin{matrix} 6 x^{2} - 15 a x & = 2 b x - 5 a b \\ 6 x^{2} - 2 b x - 15 a x + 5 a b & = 0 \\ 2 x (3 x - b) - 5 a (3 x - b) & = 0 \\ (2 x - 5 a) (3 x - b) & = 0 \\ 3 x - b & = 0 \\ 3 x & = b \\ x_{1} & = \frac{b}{3} \\ 2 x - 5 a & = 0 \\ 2 x & = 5 a \\ x_{2} & = \frac{5 a}{2} \end{matrix}$
$\begin{matrix} \frac{3 x}{4} + \frac{a}{2} - \frac{x^{2}}{2 a} & = 0 \\ M u l t i p l i c a n d o l a e c u a c i ó n p o r - 4 a \\ 2 x^{2} - 3 a x - 2 a^{2} & = 0 \\ 2 x^{2} + a x - 4 a x - 2 a^{2} & = 0 \\ x (2 x + a) - 2 a (2 x + a) & = 0 \\ (x - 2 a) (2 x + a) & = 0 \\ 2 x + a & = 0 \\ 2 x & = - a \\ x_{1} & = - \frac{a}{2} \\ x - 2 a & = 0 \\ x_{2} & = 2 a \end{matrix}$
$\begin{matrix} \frac{2 x - b}{2} & = \frac{2 b x - b^{2}}{3 x} \\ 3 x (2 x - b) & = 2 (2 b x - b^{2}) \\ 6 x^{2} - 3 b x & = 4 b x - 2 b^{2} \\ 6 x^{2} - 3 b x - 4 b x + 2 b^{2} & = 0 \\ 6 x^{2} - 7 b x + 2 b^{2} & = 0 \\ 6 x^{2} - 3 b x - 4 b x + 2 b^{2} & = 0 \\ 3 x (2 x - b) - 2 b (2 x - b) & = 0 \\ (3 x - 2 b) (2 x - b) & = 0 \\ 2 x - b & = 0 \\ 2 x & = b \\ x_{1} & = \frac{b}{2} \\ 3 x - 2 b & = 0 \\ 3 x & = 2 b \\ x_{2} & = \frac{2 b}{3} \end{matrix}$
$\begin{matrix} \frac{a + x}{a - x} + \frac{a - 2 x}{a + x} & = - 4 \\ \frac{{(a + x)}^{2} + (a - x) (a - 2 x)}{(a - x) (a + x)} & = - 4 \\ \frac{a^{2} + 2 a x + x^{2} + a^{2} - 2 a x - a x + 2 x^{2}}{a^{2} - x^{2}} & = - 4 \\ 3 x^{2} - a x + 2 a^{2} & = - 4 (a^{2} - x^{2}) \\ 3 x^{2} - a x + 2 a^{2} & = - 4 a^{2} + 4 x^{2} \\ 3 x^{2} - a x + 2 a^{2} + 4 a^{2} - 4 x^{2} & = 0 \\ - x^{2} - a x + 6 a^{2} & = 0 \\ x^{2} + a x - 6 a^{2} & = 0 \\ x^{2} - 2 a x + 3 a x - 6 a^{2} & = 0 \\ x (x - 2 a) + 3 a (x - 2 a) & = 0 \\ (x - 2 a) (x + 3 a) & = 0 \\ x - 2 a & = 0 \\ x_{1} & = 2 a \\ x + 3 a & = 0 \\ x_{2} & = - 3 a \end{matrix}$
$\begin{matrix} \frac{x^{2}}{x - 1} & = \frac{a^{2}}{2 (a - 2)} \\ 2 x^{2} (a - 2) & = a^{2} (x - 1) \\ 2 a x^{2} - 4 x^{2} & = a^{2} x - a^{2} \\ 2 a x^{2} - 4 x^{2} - a^{2} x + a^{2} & = 0 \\ 2 a x^{2} - a^{2} x + a^{2} - 4 x^{2} & = 0 \\ a x (2 x - a) + (a + 2 x) (a - 2 x) & = 0 \\ - a x (a - 2 x) + (a + 2 x) (a - 2 x) & = 0 \\ (a - 2 x) (- a x + a + 2 x) & = 0 \\ a - 2 x & = 0 \\ a & = 2 x \\ x_{1} & = \frac{a}{2} \\ - a x + a + 2 x & = 0 \\ x (2 - a) & = - a \\ x_{2} & = - \frac{a}{2 - a} \leftrightarrow \frac{a}{a - 2} \end{matrix}$
$\begin{matrix} x + \frac{2}{x} & = \frac{1}{a} + 2 a \\ \frac{x^{2} + 2}{x} & = \frac{1 + 2 a^{2}}{a} \\ a (x^{2} + 2) & = x (1 + 2 a^{2}) \\ a x^{2} + 2 a & = x + 2 a^{2} x \\ a x^{2} + 2 a - x - 2 a^{2} x & = 0 \\ a x^{2} - x - 2 a^{2} x + 2 a & = 0 \\ x (a x - 1) - 2 a (a x - 1) & = 0 \\ (x - 2 a) (a x - 1) & = 0 \\ x - 2 a & = 0 \\ x_{1} & = 2 a \\ a x - 1 & = 0 \\ a x & = 1 \\ x_{2} & = \frac{1}{a} \end{matrix}$
$\begin{matrix} \frac{2 x - b}{b} - \frac{x}{x + b} & = \frac{2 x}{4 b} \\ \frac{(2 x - b) (x + b) - b x}{b (x + b)} & = \frac{2 x}{b} \\ \frac{2 x^{2} + 2 b x - b x - b^{2} - b x}{x + b} & = \frac{x}{2} \\ 2 (2 x^{2} - b^{2}) & = x (x + b) \\ 4 x^{2} - 2 b^{2} & = x^{2} + b x \\ 4 x^{2} - 2 b^{2} - x^{2} - b x & = 0 \\ 3 x^{2} - b x - 2 b^{2} & = 0 \\ 3 x^{2} - 3 b x + 2 b x - 2 b^{2} & = 0 \\ 3 x (x - b) + 2 b (x - b) & = 0 \\ (3 x + 2 b) (x - b) & = 0 \\ x - b & = 0 \\ x_{1} & = b \\ 3 x + 2 b & = 0 \\ 3 x & = - 2 b \\ x_{2} & = - \frac{2 b}{3} \end{matrix}$