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CAPITULO XXVIII

Problemas sobre ecuaciones simultaneas

Ejercicio 208

Elevar al cuadrado

$\begin{matrix} x^{2} - 2 x + 1 & = {([x^{2} - 2 x] + 1)}^{2} \\ = {(x^{2} - 2 x)}^{2} + 2 (x^{2} - 2 x) + 1 \\ = x^{4} - 4 x^{3} + 4 x^{2} + 2 x^{2} - 4 x + 1 \\ = x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1 \end{matrix}$
$\begin{matrix} 2 x^{2} + x + 1 & = {([2 x^{2} + x] + 1)}^{2} \\ = {(2 x^{2} + x)}^{2} + 2 (2 x^{2} + x) + 1 \\ = 4 x^{4} + 4 x^{3} + x^{2} + 4 x^{2} + 2 x + 1 \\ = 4 x^{4} + 4 x^{3} + 5 x^{2} + 2 x + 1 \end{matrix}$
$\begin{matrix} x^{2} - 5 x + 2 & = {([x^{2} - 5 x] + 2)}^{2} \\ = {(x^{2} - 5 x)}^{2} + 4 (x^{2} - 5 x) + 2^{2} \\ = x^{4} - 10 x^{3} + 25 x^{2} + 4 x^{2} - 20 x + 4 \\ = x^{4} - 10 x^{3} + 29 x^{2} - 20 x + 4 \end{matrix}$
$\begin{matrix} x^{3} - 5 x^{2} + 6 & = {([x^{3} - 5 x^{2}] + 6)}^{2} \\ = {(x^{3} - 5 x^{2})}^{2} + 12 (x^{3} - 5 x^{2}) + 6^{2} \\ = x^{6} - 10 x^{5} + 25 x^{4} + 12 x^{3} - 60 x^{2} + 36 \end{matrix}$
$\begin{matrix} 4 a^{4} - 3 a^{2} + 5 & = {([4 a^{4} - 3 a^{2}] + 5)}^{2} \\ = {(4 a^{4} - 3 a^{2})}^{2} + 10 (4 a^{4} - 3 a^{2}) + 5^{2} \\ = 16 a^{8} - 24 a^{6} + 9 a^{4} + 40 a^{4} - 30 a^{2} + 25 \\ = 16 a^{8} - 24 a^{6} + 49 a^{4} - 30 a^{2} + 25 \end{matrix}$
$\begin{matrix} x + 2 y - z & = {([x + 2 y] - z)}^{2} \\ = {(x + 2 y)}^{2} - 2 z (x + 2 y) + z^{2} \\ = x^{2} + 4 x y + 4 y^{2} - 2 x z - 4 y z + z^{2} \end{matrix}$
$\begin{matrix} 3 - x^{3} - x^{6} & = {([3 - x^{3}] - x^{6})}^{2} \\ = {(3 - x^{3})}^{2} - 2 x^{6} (3 - x^{3}) + x^{12} \\ = 9 - 6 x^{3} + x^{6} - 6 x^{6} + 2 x^{9} + x^{12} \\ = 9 - 6 x^{3} - 5 x^{6} + 2 x^{9} + x^{12} \end{matrix}$
$\begin{matrix} 5 x^{4} - 7 x^{2} + 3 x & = {([5 x^{4} - 7 x^{2}] + 3 x)}^{2} \\ = {(5 x^{4} - 7 x^{2})}^{2} + 6 x (5 x^{4} - 7 x^{2}) + {(3 x)}^{2} \\ = 25 x^{8} - 70 x^{6} + 49 x^{4} + 30 x^{5} - 42 x^{3} + 9 x^{2} \\ = 25 x^{8} - 70 x^{6} + 30 x^{5} + 49 x^{4} - 42 x^{3} + 9 x^{2} \end{matrix}$
$\begin{matrix} 2 a^{2} + 2 a b - 3 b^{2} & = {([2 a^{2} + 2 a b] - 3 b^{2})}^{2} \\ = {(2 a^{2} + 2 a b)}^{2} - 6 b^{2} (2 a^{2} + 2 a b) + {(- 3 b^{2})}^{2} \\ = 4 a^{4} + 8 a^{3} b + 4 a^{2} b^{2} - 12 a^{2} b^{2} - 12 a b^{3} + 9 b^{4} \\ = 4 a^{4} + 8 a^{3} b - 8 a^{2} b^{2} - 12 a b^{3} + 9 b^{4} \end{matrix}$
$\begin{matrix} m^{3} - 2 m^{2} n + 2 n^{4} & = {([m^{3} - 2 m^{2} n] + 2 n^{4})}^{2} \\ = {(m^{3} - 2 m^{2} n)}^{2} + 4 n^{4} (m^{3} - 2 m^{2} n) + {(2 n^{4})}^{2} \\ = m^{6} - 4 m^{5} n + 4 m^{4} n^{2} + 4 m^{3} n^{4} - 8 m^{2} n^{5} + 4 n^{8} \end{matrix}$
$\begin{matrix} \frac{a}{2} - b + \frac{c}{4} & = {([\frac{a}{2} - b] + \frac{c}{4})}^{2} \\ = {(\frac{a}{2} - b)}^{2} + \frac{c}{2} (\frac{a}{2} - b) + {(\frac{c}{4})}^{2} \\ = \frac{a^{2}}{4} - a b + b^{2} + \frac{a c}{4} - \frac{b c}{2} + \frac{c^{2}}{16} \\ = \frac{a^{2}}{4} + b^{2} + \frac{c^{2}}{16} - a b + \frac{a c}{4} - \frac{b c}{2} \end{matrix}$
$\begin{matrix} \frac{x}{5} - 5 y + \frac{5}{3} & = {([\frac{x}{5} - 5 y] + \frac{5}{3})}^{2} \\ = {(\frac{x}{5} - 5 y)}^{2} + \frac{10}{3} (\frac{x}{5} - 5 y) + {(\frac{5}{3})}^{2} \\ = \frac{x^{2}}{25} - 2 x y + 25 y^{2} + \frac{2}{3} x - \frac{50}{3} y + \frac{25}{9} \end{matrix}$
$\begin{matrix} \frac{1}{2} x^{2} - x + \frac{2}{3} & = {([\frac{1}{2} x^{2} - x] + \frac{2}{3})}^{2} \\ = {(\frac{1}{2} x^{2} - x)}^{2} + \frac{4}{3} (\frac{1}{2} x^{2} - x) + {(\frac{2}{3})}^{2} \\ = \frac{x^{4}}{4} - x^{3} + x^{2} + \frac{2}{3} x^{2} - \frac{4}{3} x + \frac{4}{9} \\ = \frac{x^{4}}{4} - x^{3} + \frac{5}{3} x^{2} - \frac{4}{3} x + \frac{4}{9} \end{matrix}$
$\begin{matrix} \frac{a}{x} - \frac{1}{3} + \frac{x}{a} & = {([\frac{a}{x} - \frac{1}{3}] + \frac{x}{a})}^{2} \\ = {(\frac{a}{x} - \frac{1}{3})}^{2} + \frac{2 x}{a} (\frac{a}{x} - \frac{1}{3}) + {(\frac{x}{a})}^{2} \\ = \frac{a^{2}}{x^{2}} - \frac{2 a}{3 x} + \frac{1}{9} + 2 - \frac{2 x}{3 a} + \frac{x^{2}}{a^{2}} \\ = \frac{a^{2}}{x^{2}} - \frac{2 a}{3 x} - \frac{2 x}{3 a} + \frac{x^{2}}{a^{2}} + \frac{19}{9} \end{matrix}$
$\begin{matrix} \frac{3}{4} a^{2} - \frac{1}{2} a + \frac{4}{5} & = {([\frac{3}{4} a^{2} - \frac{1}{2} a] + \frac{4}{5})}^{2} \\ = {(\frac{3}{4} a^{2} - \frac{1}{2} a)}^{2} + \frac{8}{3} (\frac{3}{4} a^{2} - \frac{1}{2} a) + {(\frac{4}{5})}^{2} \\ = \frac{9}{16} a^{4} - \frac{3}{4} a^{3} + \frac{1}{4} a^{2} + 2 a^{2} - \frac{4}{3} a + \frac{16}{25} \\ = \frac{9}{16} a^{4} - \frac{3}{4} a^{3} + \frac{29}{20} a^{2} - \frac{4}{3} a + \frac{16}{25} \end{matrix}$
$\begin{matrix} \frac{a^{2}}{4} - \frac{3}{5} + \frac{b^{2}}{9} & = {([\frac{a^{2}}{4} - \frac{3}{5}] + \frac{b^{2}}{9})}^{2} \\ = {(\frac{a^{2}}{4} - \frac{3}{5})}^{2} + \frac{2 b^{2}}{9} (\frac{a^{2}}{4} - \frac{3}{5}) + {(\frac{b^{2}}{9})}^{2} \\ = \frac{a^{4}}{16} - \frac{3}{10} a^{2} + \frac{9}{25} + \frac{a^{2} b^{2}}{18} - \frac{2 b^{2}}{15} + \frac{b^{4}}{81} \\ = \frac{a^{4}}{16} - \frac{3}{10} a^{2} + \frac{a^{2} b^{2}}{18} - \frac{2 b^{2}}{15} + \frac{b^{4}}{81} + \frac{9}{25} \end{matrix}$
$\begin{matrix} x^{3} - x^{2} + x + 1 & = {([x^{3} - x^{2}] + [x + 1])}^{2} \\ = {(x^{3} - x^{2})}^{2} + 2 (x^{3} - x^{2}) (x + 1) + {(x + 1)}^{2} \\ = x^{6} - 2 x^{5} + x^{4} + 2 (x^{4} + x^{3} - x^{3} - x^{2}) + x^{2} + 2 x + 1 \\ = x^{6} - 2 x^{5} + x^{4} + 2 x^{4} - 2 x^{2} + x^{2} + 2 x + 1 \\ = x^{6} - 2 x^{5} + 3 x^{4} - x^{2} + 2 x + 1 \end{matrix}$
$\begin{matrix} x^{3} - 3 x^{2} - 2 x + 2 & = {([x^{3} - 3 x^{2}] - [2 x - 2])}^{2} \\ = {(x^{3} - 3 x^{2})}^{2} - 2 (x^{3} - 3 x^{2}) (2 x - 2) + {(2 x - 2)}^{2} \\ = x^{6} - 6 x^{5} + 9 x^{4} - 2 (2 x^{4} - 2 x^{3} - 6 x^{3} + 6 x^{2}) + 4 x^{2} - 8 x + 4 \\ = x^{6} - 6 x^{5} + 9 x^{4} - 4 x^{4} + 16 x^{3} - 12 x^{2} + 4 x^{2} - 8 x + 4 \\ = x^{6} - 6 x^{5} + 5 x^{4} + 16 x^{3} - 8 x^{2} - 8 x + 4 \end{matrix}$
$\begin{matrix} x^{4} + 3 x^{2} - 4 x + 5 & = {([x^{4} + 3 x^{2}] - [4 x - 5])}^{2} \\ = {(x^{4} + 3 x^{2})}^{2} - 2 (x^{4} + 3 x^{2}) (4 x - 5) + {(4 x - 5)}^{2} \\ = x^{8} + 6 x^{6} + 9 x^{4} - 2 (4 x^{5} - 5 x^{4} + 12 x^{3} - 15 x^{2}) + 16 x^{2} - 40 x + 25 \\ = x^{8} + 6 x^{6} + 9 x^{4} - 8 x^{5} + 10 x^{4} - 24 x^{3} + 30 x^{2} + 16 x^{2} - 40 x + 25 \\ = x^{8} + 6 x^{6} - 8 x^{5} + 19 x^{4} - 24 x^{3} + 46 x^{2} - 40 x + 25 \end{matrix}$
$\begin{matrix} x^{4} - 4 x^{3} + 2 x - 3 & = {([x^{4} - 4 x^{3}] + [2 x - 3])}^{2} \\ = {(x^{4} - 4 x^{3})}^{2} + 2 (x^{4} - 4 x^{3}) (2 x - 3) + {(2 x - 3)}^{2} \\ = x^{8} - 8 x^{7} + 16 x^{6} + 2 (2 x^{5} - 3 x^{4} - 8 x^{4} + 12 x^{3}) + 4 x^{2} - 12 x + 9 \\ = x^{8} - 8 x^{7} + 16 x^{6} + 4 x^{5} - 22 x^{4} + 24 x^{3} + 4 x^{2} - 12 x + 9 \end{matrix}$
$\begin{matrix} 3 - 6 a + a^{2} - a^{3} & = {([3 - 6 a] + [a^{2} - a^{3}])}^{2} \\ = {(3 - 6 a)}^{2} + 2 (3 - 6 a) (a^{2} - a^{3}) + {(a^{2} - a^{3})}^{2} \\ = 9 - 36 a + 36 a^{2} + 2 (3 a^{2} - 3 a^{3} - 6 a^{3} + 6 a^{4}) + a^{4} - 2 a^{5} + a^{6} \\ = 9 - 36 a + 36 a^{2} + 6 a^{2} - 18 a^{3} + 12 a^{4} + a^{4} - 2 a^{5} + a^{6} \\ = 9 - 36 a + 42 a^{2} - 18 a^{3} + 13 a^{4} - 2 a^{5} + a^{6} \end{matrix}$
$\begin{matrix} \frac{1}{2} x^{3} - x^{2} + \frac{2}{3} x + 2 & = {([\frac{1}{2} x^{3} - x^{2}] + [\frac{2}{3} x + 2])}^{2} \\ = {(\frac{1}{2} x^{3} - x^{2})}^{2} + 2 (\frac{1}{2} x^{3} - x^{2}) (\frac{2}{3} x + 2) + {(\frac{2}{3} x + 2)}^{2} \\ = \frac{x^{6}}{4} - x^{5} + x^{4} + 2 (\frac{1}{3} x^{4} + x^{3} - \frac{2}{3} x^{3} - 2 x^{2}) + \frac{4}{9} x^{2} + \frac{8}{3} x + 4 \\ = \frac{x^{6}}{4} - x^{5} + x^{4} + \frac{2}{3} x^{4} + \frac{2}{3} x^{3} - 4 x^{2} + \frac{4}{9} x^{2} + \frac{8}{3} x + 4 \\ = \frac{x^{6}}{4} - x^{5} + \frac{5}{3} x^{4} + \frac{2}{3} x^{3} - \frac{32}{9} x^{2} + \frac{8}{3} x + 4 \end{matrix}$
$\begin{matrix} \frac{1}{2} a^{3} - \frac{2}{3} a^{2} + \frac{3}{4} a - \frac{1}{2} & = {([\frac{1}{2} a^{3} - \frac{2}{3} a^{2}] + [\frac{3}{4} a - \frac{1}{2}])}^{2} \\ = {(\frac{1}{2} a^{3} - \frac{2}{3} a^{2})}^{2} + 2 (\frac{1}{2} a^{3} - \frac{2}{3} a^{2}) (\frac{3}{4} a - \frac{1}{2}) + {(\frac{3}{4} a - \frac{1}{2})}^{2} \\ = \frac{a^{6}}{4} - \frac{2 a^{5}}{3} + \frac{4 a^{4}}{9} + 2 (\frac{3 a^{4}}{8} - \frac{a^{3}}{4} - \frac{a^{3}}{2} + \frac{a^{2}}{3}) + \frac{9 a^{2}}{16} - \frac{3 a}{4} + \frac{1}{4} \\ = \frac{a^{6}}{4} - \frac{2 a^{5}}{3} + \frac{4 a^{4}}{9} + \frac{3 a^{4}}{4} - \frac{3 a^{3}}{2} + \frac{2 a^{2}}{3} + \frac{9 a^{2}}{16} - \frac{3 a}{4} + \frac{1}{4} \\ = \frac{a^{6}}{4} - \frac{2 a^{5}}{3} + \frac{43 a^{4}}{36} - \frac{3 a^{3}}{2} + \frac{59 a^{2}}{48} - \frac{3 a}{4} + \frac{1}{4} \end{matrix}$
$\begin{matrix} x^{5} - x^{4} + x^{3} - x^{2} + x - 2 & = {([x^{5} - x^{4} + x^{3}] - [x^{2} - x + 2])}^{2} \\ = {(x^{5} - x^{4} + x^{3})}^{2} - 2 (x^{5} - x^{4} + x^{3}) (x^{2} - x + 2) + {(x^{2} - x + 2)}^{2} \\ = {([x^{5} - x^{4}] + x^{3})}^{2} - 2 (x^{7} - x^{6} + 2 x^{5} - x^{6} + x^{5} - 2 x^{4} + x^{5} - x^{4} + 2 x^{3}) + {([x^{2} - x] + 2)}^{2} \\ = {(x^{5} - x^{4})}^{2} + 2 x^{3} (x^{5} - x^{4}) + x^{6} - 2 (x^{7} - 2 x^{6} + 4 x^{5} - 3 x^{4} + 2 x^{3}) + {(x^{2} - x)}^{2} + 4 (x^{2} - x) + 2^{2} \\ = x^{10} - 2 x^{9} + x^{8} + 2 x^{8} - 2 x^{7} + x^{6} - 2 x^{7} + 4 x^{6} - 8 x^{5} + 6 x^{4} - 4 x^{3} + x^{4} - 2 x^{3} + x^{2} + 4 x^{2} - 4 x + 4 \\ = x^{10} - 2 x^{9} + 3 x^{8} - 4 x^{7} + 5 x^{6} - 8 x^{5} + 7 x^{4} - 6 x^{3} + 5 x^{2} - 4 x + 4 \end{matrix}$