Ejercicio 284 – Solucionario Baldor

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DESCARGA

CAPITULO XXXVI

ECUACIONES BINOMIAS Y TRINOMIAS

Ejercicio 284

Resolver las ecuaciones:

$\begin{matrix} x^{6} - 7 x^{3} - 8 & = 0 \\ x^{6} - 8 x^{3} + x^{3} - 8 & = 0 \\ x^{3} (x^{3} - 8) + (x^{3} - 8) & = 0 \\ (x^{3} - 8) (x^{3} + 1) & = 0 \\ (x - 2) (x^{2} + 2 x + 4) (x + 1) (x^{2} - x + 1) & = 0 \\ x_{1} - 2 & = 0 \\ x_{1} & = 2 \\ x_{2} + 1 & = 0 \\ x_{2} & = - 2 \\ x^{2} + 2 x + 4 & = 0 \\ x & = \frac{- 2 \pm \sqrt{2^{2} - 4 (1) (4)}}{2 (1)} \\ x & = \frac{- 2 \pm \sqrt{4 - 16}}{2} \\ x & = \frac{- 2 \pm \sqrt{- 12}}{2} \\ x & = \frac{- 2 \pm \sqrt{(- 1) (3) (2^{2})}}{2} \\ x & = \frac{- 2 \pm 2 \sqrt{- 1} \sqrt{3}}{2} \\ x & = \frac{2 (- 1 \pm \sqrt{3} i)}{2} \\ x_{3} & = - 1 + \sqrt{3} i \\ x_{4} & = - 1 - \sqrt{3} i \\ x^{2} - x + 1 & = 0 \\ x & = \frac{- (- 1) \pm \sqrt{{(- 1)}^{2} - 4 (1) (1)}}{2 (1)} \\ x & = \frac{1 \pm \sqrt{1 - 4}}{2} \\ x & = \frac{1 \pm \sqrt{- 1} \sqrt{3}}{2} \\ x & = \frac{1 \pm \sqrt{3} i}{2} \\ x_{5} & = \frac{1 + \sqrt{3} i}{2} \\ x_{6} & = \frac{1 - \sqrt{3} i}{2} \end{matrix}$
$\begin{matrix} x^{6} + 30 x^{3} + 81 & = 0 \\ x^{6} + 27 x^{3} + 3 x^{3} + 81 & = 0 \\ x^{3} (x^{3} + 27) + 3 (x^{3} + 27) & = 0 \\ (x^{3} + 3) (x^{3} + 27) & = 0 \\ (x^{3} + 3) (x + 3) (x^{2} - 3 x + 9) & = 0 \\ x_{1}^{3} + 3 & = 0 \\ x_{1}^{3} & = - 3 \\ x_{1} & = \sqrt[3]{- 3} \\ x_{1} & = - \sqrt[3]{3} \\ x_{2} + 3 & = 0 \\ x_{2} & = - 3 \\ x^{2} - 3 x + 9 & = 0 \\ x & = \frac{- (- 3) \pm \sqrt{{(- 3)}^{2} - 4 (1) (9)}}{2 (1)} \\ x & = \frac{3 \pm \sqrt{9 - 36}}{2} \\ x & = \frac{3 \pm \sqrt{- 27}}{2} \\ x & = \frac{3 \pm \sqrt{(- 1) (3^{3})}}{2} \\ x & = \frac{3 \pm 3 \sqrt{3} \sqrt{- 1}}{2} \\ x & = \frac{3 \pm 3 \sqrt{3} i}{2} \\ x_{3} & = \frac{3 - 3 \sqrt{3} i}{2} \\ x_{4} & = \frac{3 + 3 \sqrt{3} i}{2} \end{matrix}$
$\begin{matrix} 8 x^{6} + 15 x^{3} - 2 & = 0 \\ 8 x^{6} + 16 x^{3} - x^{3} - 2 & = 0 \\ 8 x^{3} (x^{2} + 2) - (x^{3} + 2) & = 0 \\ (8 x^{3} - 1) (x^{3} + 2) & = 0 \\ 8 x_{1}^{3} - 1 & = 0 \\ 8 x_{1}^{3} & = 1 \\ x_{1}^{3} & = \frac{1}{8} \\ x_{1} & = \sqrt{\frac{1}{8}} \\ x_{1} & = \frac{1}{2} \\ x_{2}^{3} + 2 & = 0 \\ x_{2}^{3} & = - 2 \\ x_{2} & = \sqrt[3]{- 2} \\ x_{2} & = - \sqrt[3]{2} \end{matrix}$
$\begin{matrix} x^{8} - 41 x^{4} + 400 & = 0 \\ x^{8} - 16 x^{4} - 25 x^{4} + 400 & = 0 \\ x^{4} (x^{4} - 16) - 25 (x^{4} - 16) & = 0 \\ (x^{4} - 25) (x^{4} - 16) & = 0 \\ (x^{2} + 5) (x^{2} - 5) (x^{2} - 4) (x^{2} + 4) & = 0 \\ (x^{2} + 5) (x^{2} - 5) (x^{2} + 4) (x + 2) (x - 2) & = 0 \\ x_{1} + 2 & = 0 \\ x_{1} & = - 2 \\ x_{2} - 2 & = 0 \\ x_{2} & = 2 \\ x^{2} + 5 & = 0 \\ x^{2} & = - 5 \\ x & = \pm \sqrt{- 5} \\ x_{3} & = \sqrt{5} i \\ x_{4} & = - \sqrt{5} i \\ x^{2} - 5 & = 0 \\ x^{2} & = 5 \\ x & = \pm \sqrt{5} \\ x_{5} & = \sqrt{5} \\ x_{6} & = - \sqrt{5} \\ x^{2} + 4 & = 0 \\ x^{2} & = - 4 \\ x & = \pm \sqrt{- 4} \\ x & = \pm \sqrt{(- 1) (2^{2})} \\ x & = \pm 2 \sqrt{- 1} \\ x & = \pm 2 i \\ x_{7} & = 2 i \\ x_{8} & = - 2 i \end{matrix}$
$\begin{matrix} x^{10} - 33 x^{5} + 32 & = 0 \\ x^{10} - x^{5} - 32 x^{5} + 32 & = 0 \\ x^{5} (x^{2} - 1) - 32 (x^{5} - 1) & = 0 \\ (x^{5} - 32) (x^{5} - 1) & = 0 \\ x^{5} - 32 & = 0 \\ x^{5} & = 32 \\ x_{1} & = \sqrt[5]{32} \\ x_{1} & = 2 \\ x^{5} - 1 & = 0 \\ x^{5} & = 1 \\ x & = \sqrt[5]{1} \\ x_{2} & = 1 \end{matrix}$
$\begin{matrix} x^{- 4} - 13 x^{- 2} + 36 & = 0 \\ (x^{- 2} - 9) (x^{- 2} - 4) & = 0 \\ x^{- 2} - 9 & = 0 \\ x^{- 2} & = 9 \\ \frac{1}{x^{2}} & = 9 \\ x^{2} & = \frac{1}{9} \\ x & = \pm \sqrt{\frac{1}{9}} \\ x & = \pm \frac{1}{3} \\ x_{1} & = \frac{1}{3} \\ x_{2} & = - \frac{1}{3} \\ x^{- 2} - 4 & = 0 \\ x^{- 2} & = 4 \\ \frac{1}{x^{2}} & = 4 \\ x^{2} & = \frac{1}{4} \\ x & = \pm \sqrt{\frac{1}{4}} \\ x & = \pm \frac{1}{2} \\ x_{3} & = \frac{1}{2} \\ x_{4} & = - \frac{1}{2} \end{matrix}$
$\begin{matrix} x^{- 6} + 35 x^{- 3} & = - 216 \\ x^{- 6} + 35 x^{- 3} + 216 & = 0 \\ x^{- 6} + 27 x^{- 3} + 8 x^{- 3} + 216 & = 0 \\ x^{- 3} (x^{- 3} + 27) + 8 (x^{- 3} + 27) & = 0 \\ (x^{- 3} + 8) (x^{- 3} + 27) & = 0 \\ x^{- 3} + 8 & = 0 \\ x^{- 3} & = - 8 \\ \frac{1}{x^{3}} & = - 8 \\ x^{3} & = - \frac{1}{8} \\ x_{1} & = \sqrt[3]{- \frac{1}{8}} \\ x_{1} & = - \frac{1}{2} \\ x^{- 3} + 27 & = 0 \\ x^{- 3} & = - 27 \\ \frac{1}{x^{3}} & = - 27 \\ x^{3} & = - \frac{1}{27} \\ x_{2} & = \sqrt[3]{- \frac{1}{27}} \\ x_{2} & = - \frac{1}{3} \end{matrix}$
$\begin{matrix} x^{- 10} & = 242 x^{- 5} + 243 \\ x^{- 10} - 242 x^{- 5} - 243 & = 0 \\ (x^{- 5} - 243) (x^{- 5} + 1) & = 0 \\ x^{- 5} - 243 & = 0 \\ x^{- 5} & = 243 \\ \frac{1}{x^{5}} & = 243 \\ x^{5} & = \frac{1}{243} \\ x_{1} & = \sqrt[5]{\frac{1}{243}} \\ x_{1} & = \frac{1}{3} \\ x^{- 5} + 1 & = 0 \\ x^{- 5} & = - 1 \\ \frac{1}{x^{5}} & = - 1 \\ x^{5} & = - 1 \\ x_{2} & = \sqrt[5]{- 1} \\ x_{2} & = - 1 \end{matrix}$
$\begin{matrix} x^{3} - 9 x^{\frac{3}{2}} + 8 & = 0 \\ x^{3} - x^{\frac{3}{2}} - 8 x^{\frac{3}{2}} + 8 & = 0 \\ x^{\frac{3}{2}} (x^{\frac{3}{2}} - 1) - 8 (x^{\frac{3}{2}} - 1) & = 0 \\ (x^{\frac{3}{2}} - 8) (x^{\frac{3}{2}} - 1) & = 0 \\ x^{\frac{3}{2}} - 8 & = 0 \\ x^{\frac{3}{2}} & = 8 \\ {(x^{\frac{3}{2}})}^{\frac{2}{3}} & = 8^{\frac{2}{3}} \\ x_{1} & = \sqrt[3]{8^{2}} \\ x_{1} & = \sqrt[3]{64} \\ x_{1} & = 4 \\ x^{\frac{3}{2}} - 1 & = 0 \\ {(x^{\frac{3}{2}})}^{\frac{2}{3}} & = 1^{\frac{2}{3}} \\ x_{2} & = 1 \end{matrix}$
$\begin{matrix} x + x^{\frac{1}{2}} & = 6 \\ x - 2 x^{\frac{1}{2}} + 3 x^{\frac{1}{2}} - 6 & = 0 \\ x^{\frac{1}{2}} (x^{\frac{1}{2}} - 2) + 3 (x^{\frac{1}{2}} - 2) & = 0 \\ (x^{\frac{1}{2}} + 3) (x^{\frac{1}{2}} - 2) & = 0 \\ x^{\frac{1}{2}} + 3 & = 0 \\ x^{\frac{1}{2}} & = - 3 \\ {(x^{\frac{1}{2}})}^{2} & = {(- 3)}^{2} \\ x_{1} & = 9 \\ x^{\frac{1}{2}} - 2 & = 0 \\ x^{\frac{1}{2}} & = 2 \\ {(x^{\frac{1}{2}})}^{2} & = 2^{2} \\ x_{2} & = 4 \end{matrix}$
$\begin{matrix} 3 x & = 16 \sqrt{x} - 5 \\ 3 x - 16 x^{\frac{1}{2}} + 5 & = 0 \\ 3 x - x^{\frac{1}{2}} - 15 x^{\frac{1}{2}} + 5 & = 0 \\ x^{\frac{1}{2}} (3 x^{\frac{1}{2}} - 1) - 5 (3 x^{\frac{1}{2}} - 1) & = 0 \\ (x^{\frac{1}{2}} - 5) (3 x^{\frac{1}{2}} - 1) & = 0 \\ x^{\frac{1}{2}} - 5 & = 0 \\ x^{\frac{1}{2}} & = 5 \\ {(x^{\frac{1}{2}})}^{2} & = 5^{2} \\ x_{1} & = 25 \\ 3 x^{\frac{1}{2}} - 1 & = 0 \\ 3 x^{\frac{1}{2}} & = 1 \\ x^{\frac{1}{2}} & = \frac{1}{3} \\ {(x^{\frac{1}{2}})}^{2} & = {(\frac{1}{3})}^{2} \\ x & = \frac{1}{9} \end{matrix}$
$\begin{matrix} 2 x^{\frac{1}{2}} - 5 x^{\frac{1}{4}} + 2 & = 0 \\ 2 x^{\frac{1}{2}} - x^{\frac{1}{4}} - 4 x^{\frac{1}{4}} + 2 & = 0 \\ x^{\frac{1}{4}} (2 x^{\frac{1}{4}} - 1) - 2 (2 x^{\frac{1}{4}} - 1) & = 0 \\ (x^{\frac{1}{4}} - 2) (2 x^{\frac{1}{4}} - 1) & = 0 \\ x^{\frac{1}{4}} - 2 & = 0 \\ x^{\frac{1}{4}} & = 2 \\ {(x^{\frac{1}{4}})}^{4} & = 2^{4} \\ x_{1} & = 16 \\ 2 x^{\frac{1}{4}} - 1 & = 0 \\ 2 x^{\frac{1}{4}} & = 1 \\ x^{\frac{1}{4}} & = \frac{1}{2} \\ {(x^{\frac{1}{4}})}^{4} & = {(\frac{1}{2})}^{4} \\ x_{2} & = \frac{1}{16} \end{matrix}$