Ejercicio 283 – Solucionario Baldor

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CAPITULO XXXVI

ECUACIONES BINOMIAS Y TRINOMIAS

Ejercicio 283

Resolver las ecuaciones siguientes, hallando todas las raíces:

$\begin{matrix} x^{4} - 10 x^{2} + 9 & = 0 \\ (x^{2} - 9) (x^{2} - 1) & = 0 \\ (x - 3) (x + 3) (x - 1) (x + 1) & = 0 \\ x_{1} - 3 & = 0 \\ x_{1} & = 3 \\ x_{2} + 3 & = 0 \\ x_{2} & = - 3 \\ x_{3} - 1 & = 0 \\ x_{3} & = 1 \\ x_{4} + 1 & = 0 \\ x_{4} & = - 1 \end{matrix}$
$\begin{matrix} x^{4} - 13 x^{2} + 36 & = 0 \\ (x^{2} - 4) (x^{2} - 9) & = 0 \\ (x - 2) (x + 2) (x - 3) (x + 3) & = 0 \\ x_{1} - 2 & = 0 \\ x_{1} & = 2 \\ x_{2} + 2 & = 0 \\ x_{2} & = - 2 \\ x_{3} - 3 & = 0 \\ x_{3} & = 3 \\ x_{4} + 3 & = 0 \\ x_{4} & = - 3 \end{matrix}$
$\begin{matrix} x^{4} - 29 x^{2} + 100 & = 0 \\ (x^{2} - 25) (x^{2} - 4) & = 0 \\ (x - 5) (x + 5) (x - 2) (x + 2) & = 0 \\ x_{1} - 5 & = 0 \\ x_{1} & = 5 \\ x_{2} + 5 & = 0 \\ x_{2} & = - 5 \\ x_{3} - 2 & = 0 \\ x_{3} & = 2 \\ x_{4} + 2 & = 0 \\ x_{4} & = - 2 \end{matrix}$
$\begin{matrix} x^{4} - 61 x^{2} + 900 & = 0 \\ (x^{2} - 36) (x^{2} - 25) & = 0 \\ (x - 6) (x + 6) (x - 5) (x + 5) & = 0 \\ x_{1} - 6 & = 0 \\ x_{1} & = 6 \\ x_{2} + 6 & = 0 \\ x_{2} & = - 6 \\ x_{3} - 5 & = 0 \\ x_{3} & = 5 \\ x_{4} + 5 & = 0 \\ x_{4} & = - 5 \end{matrix}$
$\begin{matrix} x^{4} + 3 x^{2} - 4 & = 0 \\ (x^{2} + 4) (x^{2} - 1) & = 0 \\ (x^{2} + 4) (x - 1) (x + 1) & = 0 \\ x_{1} - 1 & = 0 \\ x_{1} & = 1 \\ x_{2} + 1 & = 0 \\ x_{2} & = - 1 \\ x^{2} + 4 & = 0 \\ x^{2} & = - 4 \\ x & = \pm \sqrt{(- 1) {(2)}^{2}} \\ x & = \pm 2 \sqrt{- 1} \\ x & = \pm 2 i \\ x_{3} & = 2 i \\ x_{4} & = - 2 i \end{matrix}$
$\begin{matrix} x^{4} + 16 x^{2} - 225 & = 0 \\ (x^{2} + 25) (x^{2} - 9) & = 0 \\ (x^{2} + 25) (x - 3) (x + 3) & = 0 \\ x_{1} - 3 & = 0 \\ x_{1} & = 3 \\ x_{2} + 3 & = 0 \\ x_{2} & = - 3 \\ x^{2} + 25 & = 0 \\ x^{2} & = - 25 \\ x & = \pm \sqrt{(- 1) (5^{2})} \\ x & = \pm 5 \sqrt{- 1} \\ x & = \pm 5 i \\ x_{3} & = 5 i \\ x_{4} & = - 5 i \end{matrix}$
$\begin{matrix} x^{4} - 45 x^{2} - 196 & = 0 \\ (x^{2} - 49) (x^{2} + 4) & = 0 \\ (x - 7) (x + 7) (x^{2} + 4) & = 0 \\ x_{1} - 7 & = 0 \\ x_{1} & = 7 \\ x_{2} + 7 & = 0 \\ x_{2} & = - 7 \\ x^{2} + 4 & = 0 \\ x^{2} & = - 4 \\ x & = \pm \sqrt{- 2^{2}} \\ x & = \pm 2 \sqrt{- 1} \\ x & = \pm 2 i \\ x_{3} & = 2 i \\ x_{4} & = - 2 i \end{matrix}$
$\begin{matrix} x^{4} - 6 x^{2} + 5 & = 0 \\ (x^{2} - 5) (x^{2} - 1) & = 0 \\ (x^{2} - 5) (x - 1) (x + 1) & = 0 \\ x_{1} - 1 & = 0 \\ x_{1} & = 1 \\ x_{2} + 1 & = 0 \\ x_{2} & = - 1 \\ x^{2} - 5 & = 0 \\ x & = \pm \sqrt{5} \\ x_{3} & = \sqrt{5} \\ x_{4} & = - \sqrt{5} \end{matrix}$
$\begin{matrix} 4 x^{4} - 37 x^{2} + 9 & = 0 \\ 4 x^{4} - x^{2} - 36 x^{2} + 9 & = 0 \\ x^{2} (4 x^{2} - 1) - 9 (4 x^{2} - 1) & = 0 \\ (x^{2} - 9) (4 x^{2} - 1) & = 0 \\ (x - 3) (x + 3) (4 x^{2} - 1) & = 0 \\ x_{1} - 3 & = 0 \\ x_{1} & = 3 \\ x_{2} + 3 & = 0 \\ x_{2} & = - 3 \\ 4 x^{2} - 1 & = 0 \\ 4 x^{2} & = 1 \\ x^{2} & = \frac{1}{4} \\ x & = \pm \sqrt{\frac{1}{4}} \\ x & = \pm \frac{1}{2} \sqrt{1} \\ x_{3} & = \frac{1}{2} \sqrt{1} \\ x_{4} & = - \frac{1}{2} \sqrt{1} \end{matrix}$
$\begin{matrix} 9 x^{4} - 40 x^{2} + 16 & = 0 \\ 9 x^{4} - 36 x^{2} - 4 x^{2} + 16 & = 0 \\ 9 x^{2} (x^{2} - 4) - 4 (x^{2} - 4) & = 0 \\ (9 x^{2} - 4) (x^{2} - 4) & = 0 \\ (9 x^{2} - 4) (x - 2) (x + 2) & = 0 \\ x_{1} - 2 & = 0 \\ x_{1} & = 2 \\ x_{2} + 2 & = 0 \\ x_{2} & = - 2 \\ 9 x^{2} - 4 & = 0 \\ 9 x^{2} & = 4 \\ x^{2} & = \frac{4}{9} \\ x & = \pm \sqrt{\frac{4}{9}} \\ x & = \pm \frac{2}{3} \\ x_{3} & = \frac{2}{3} \\ x_{4} & = - \frac{2}{3} \end{matrix}$
$\begin{matrix} 25 x^{4} + 9 x^{2} - 16 & = 0 \\ 25 x^{4} + 25 x^{2} - 16 x^{2} - 16 & = 0 \\ 25 x^{2} (x^{2} + 1) - 16 (x^{2} + 1) & = 0 \\ (25 x^{2} - 16) (x^{2} + 1) & = 0 \\ x^{2} + 1 & = 0 \\ x^{2} & = - 1 \\ x & = \pm \sqrt{- 1} \\ x_{1} & = i \\ x_{2} & = - i \\ 25 x^{2} - 16 & = 0 \\ 25 x^{2} & = 16 \\ x^{2} & = \frac{16}{25} \\ x & = \pm \sqrt{\frac{16}{25}} \\ x & = \pm \frac{4}{5} \\ x_{3} & = \frac{4}{5} \\ x_{4} & = - \frac{4}{5} \end{matrix}$
$\begin{matrix} 4 x^{4} + 11 x^{2} - 3 & = 0 \\ 4 x^{4} - x^{2} + 12 x^{2} - 3 & = 0 \\ x^{2} (4 x^{2} - 1) + 3 (4 x^{2} - 1) & = 0 \\ (x^{2} + 3) (4 x^{2} - 1) & = 0 \\ 4 x^{2} - 1 & = 0 \\ 4 x^{2} & = 1 \\ x^{2} & = \frac{1}{4} \\ x & = \pm \sqrt{\frac{1}{4}} \\ x & = \pm \frac{1}{2} \\ x_{1} & = \frac{1}{2} \\ x_{2} & = - \frac{1}{2} \\ x^{2} + 3 & = 0 \\ x^{2} & = - 3 \\ x & = \pm \sqrt{- 3} \\ x & = \pm \sqrt{- 1} \sqrt{3} \\ x & = \pm \sqrt{3} i \\ x_{3} & = \sqrt{3} i \\ x_{4} & = - \sqrt{3} i \end{matrix}$
$\begin{matrix} {(2 x^{2} + 1)}^{2} - {(x^{2} - 3)}^{2} & = 80 \\ [(2 x^{2} + 1) - (x^{2} - 3)] [(2 x^{2} + 1) + (x^{2} - 3)] & = 80 \\ [2 x^{2} + 1 - x^{2} + 3] [2 x^{2} + 1 + x^{2} - 3] & = 80 \\ (x^{2} + 4) (3 x^{2} - 2) & = 80 \\ 3 x^{4} - 2 x^{2} + 12 x^{2} - 8 & = 80 \\ 3 x^{4} + 10 x^{2} - 88 & = 0 \\ 3 x^{4} - 12 x^{2} + 22 x^{2} - 88 & = 0 \\ 3 x^{2} (x^{2} - 4) + 22 (x^{2} - 4) & = 0 \\ (3 x^{2} + 22) (x^{2} - 4) & = 0 \\ x^{2} - 4 & = 0 \\ x & = \pm \sqrt{4} \\ x & = \pm 2 \\ x_{1} & = 2 \\ x_{2} & = - 2 \\ 3 x^{2} + 22 & = 0 \\ 3 x^{2} & = - 22 \\ x^{2} & = - \frac{22}{3} \\ x & = \pm \sqrt{- \frac{22}{3}} \\ x & = \pm \sqrt{(- 1) (\frac{22}{3})} \\ x & = \pm \sqrt{\frac{22}{3}} i \\ x_{3} & = \sqrt{\frac{22}{3}} i \\ x_{4} & = - \sqrt{\frac{22}{3}} i \end{matrix}$
$\begin{matrix} x^{2} (3 x^{2} + 2) & = 4 (x^{2} - 3) + 13 \\ 3 x^{4} + 2 x^{2} & = 4 x^{2} - 12 + 13 \\ 3 x^{4} + 2 x^{2} - 4 x^{2} - 1 & = 0 \\ 3 x^{4} - 2 x^{2} - 1 & = 0 \\ 3 x^{4} - 3 x^{2} + x^{2} - 1 & = 0 \\ 3 x^{2} (x^{2} - 1) + (x^{2} - 1) & = 0 \\ (3 x^{2} + 1) (x^{2} - 1) & = 0 \\ x^{2} - 1 & = 0 \\ x^{2} & = 1 \\ x & = \pm \sqrt{1} \\ x & = \pm 1 \\ x_{1} & = 1 \\ x_{2} & = - 1 \\ 3 x^{2} + 1 & = 0 \\ 3 x^{2} & = - 1 \\ x^{2} & = - \frac{1}{3} \\ x & = \pm \sqrt{- \frac{1}{3}} \\ x & = \pm \sqrt{(- 1) (\frac{1}{3})} \\ x & = \pm \sqrt{\frac{1}{3}} i \\ x_{3} & = \sqrt{\frac{1}{3}} i \\ x_{4} & = - \sqrt{\frac{1}{3}} i \end{matrix}$