Ejercicio 273 – Solucionario Baldor

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DESCARGA

CAPITULO XXXIII

Ecuaciones con radicales que se reducen a segundo grado (soluciones extrañas)

Ejercicio 273

Resolver las ecuaciones siguientes haciendo la verificación con ambas raíces:

$\begin{matrix} x + \sqrt{4 x + 1} & = 5 \\ {(\sqrt{4 x + 1})}^{2} & = {(5 - x)}^{2} \\ 4 x + 1 & = 25 - 10 x + x^{2} \\ 0 & = - 4 x - 1 + 25 - 10 x + x^{2} \\ x^{2} - 14 x + 24 & = 0 \\ (x - 12) (x - 2) & = 0 \\ x - 2 & = 0 \\ x_{1} & = 2 \\ x - 12 & = 0 \\ x_{2} & = 12 \\ V e r i f i c a c i ó n : \\ S e a x = 2 \\ 2 + \sqrt{4 (2) + 1} & = 5 \\ 2 + \sqrt{9} & = 5 \\ 2 + 3 & = 5 \\ 5 & = 5 \Rightarrow x = 2 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = 12 \\ 12 + \sqrt{4 (12) + 1} & = 5 \\ 12 + \sqrt{25} & = 5 \\ 12 + 5 & = 5 \\ 17 & \neq 5 \Rightarrow x = 12 n o s a t i s f a c e l a e c u a c i ó n, p o r t a n t o e s u n a s o l u c i ó n e x t r a ñ a \end{matrix}$
$\begin{matrix} 2 x - \sqrt{x - 1} & = 3 x - 7 \\ - \sqrt{x - 1} & = 3 x - 2 x - 7 \\ {(- \sqrt{x - 1})}^{2} & = {(x - 7)}^{2} \\ x - 1 & = x^{2} - 14 x + 49 \\ 0 & = x^{2} - 14 x - x + 1 + 49 \\ x^{2} - 15 x + 50 & = 0 \\ (x - 10) (x - 5) & = 0 \\ x - 5 & = 0 \\ x_{1} & = 5 \\ x - 10 & = 0 \\ x_{2} & = 10 \\ V e r i f i c a c i ó n : \\ S e a x = 5 \\ 2 (5) - \sqrt{5 - 1} & = 3 (5) - 7 \\ 10 - \sqrt{4} & = 15 - 7 \\ 10 - 2 & = 8 \\ 8 & = 8 \Rightarrow x = 5 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = 10 \\ 2 (10) - \sqrt{10 - 1} & = 3 (10) - 7 \\ 20 - \sqrt{9} & = 30 - 7 \\ 20 - 3 & = 23 \\ 17 & \neq 23 \Rightarrow x = 10 n o s a t i s f a c e l a e c u a c i ó n, p o r t a n t o e s u n a s o l u c i ó n e x t r a ñ a \end{matrix}$
$\begin{matrix} \sqrt{5 x - 1} + \sqrt{x + 3} & = 4 \\ {(\sqrt{5 x - 1})}^{2} & = {(4 - \sqrt{x + 3})}^{2} \\ 5 x - 1 & = 16 - 8 \sqrt{x + 3} + x + 3 \\ 5 x - 1 - x - 3 - 16 & = - 8 \sqrt{x + 3} \\ 4 x - 20 & = - 8 \sqrt{x + 3} \\ D i v i d i e n d o l a e c u a c i ó n p a r a 4 \\ {(x - 5)}^{2} & = {(- 2 \sqrt{x + 3})}^{2} \\ x^{2} - 10 x + 25 & = 4 (x + 3) \\ x^{2} - 10 x + 25 & = 4 x + 12 \\ x^{2} - 10 x + 25 - 4 x - 12 & = 0 \\ x^{2} - 14 x + 13 & = 0 \\ (x - 13) (x - 1) & = 0 \\ x - 1 & = 0 \\ x_{1} & = 1 \\ x - 13 & = 0 \\ x_{2} & = 13 \\ V e r i f i c a c i ó n : \\ S e a x = 1 \\ \sqrt{5 (1) - 1} + \sqrt{1 + 3} & = 4 \\ \sqrt{4} + \sqrt{4} & = 4 \\ 2 + 2 & = 4 \\ 4 & = 4 \Rightarrow x = 1 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = 13 \\ \sqrt{5 (13) - 1} + \sqrt{13 + 3} & = 4 \\ \sqrt{64} + \sqrt{16} & = 4 \\ 8 + 4 & = 4 \\ 12 & \neq 4 \Rightarrow x = 12 n o s a t i s f a c e l a e c u a c i ó n, p o r t a n t o e s u n a s o l u c i ó n e x t r a ñ a \end{matrix}$
$\begin{matrix} 2 \sqrt{x} - \sqrt{x + 5} & = 1 \\ {(2 \sqrt{x} - 1)}^{2} & = {(\sqrt{x + 5})}^{2} \\ 4 x - 4 \sqrt{x} + 1 & = x + 5 \\ 4 x - x - 5 + 1 & = 4 \sqrt{x} \\ {(3 x - 4)}^{2} & = {(4 \sqrt{x})}^{2} \\ 9 x^{2} - 24 x + 16 & = 16 x \\ 9 x^{2} - 24 x + 16 - 16 x & = 0 \\ 9 x^{2} - 40 x + 16 & = 0 \\ 9 x^{2} - 36 x - 4 x + 16 & = 0 \\ 9 x (x - 4) - 4 (x - 4) & = 0 \\ (x - 4) (9 x - 4) & = 0 \\ x - 4 & = 0 \\ x_{1} & = 4 \\ 9 x - 4 & = 0 \\ 9 x & = 4 \\ x_{2} & = \frac{4}{9} \\ V e r i f i c a c i ó n : \\ S e a x = 4 \\ 2 \sqrt{4} - \sqrt{4 + 5} & = 1 \\ 2 (2) - \sqrt{9} & = 1 \\ 4 - 3 & = 1 \\ 1 & = 1 \Rightarrow x = 4 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = \frac{4}{9} \\ 2 \sqrt{\frac{4}{9}} - \sqrt{\frac{4}{9} + 5} & = 1 \\ 2 (\frac{2}{3}) - \sqrt{\frac{4 + 45}{9}} & = 1 \\ \frac{4}{3} - \sqrt{\frac{49}{9}} & = 1 \\ \frac{4}{3} - \frac{7}{3} & = 1 \\ \frac{4 - 7}{3} & = 1 \\ - 1 & \neq 1 \Rightarrow x = \frac{4}{9} n o s a t i s f a c e l a e c u a c i ó n, p o r t a n t o e s u n a s o l u c i ó n e x t r a ñ a \end{matrix}$
$\begin{matrix} \sqrt{2 x - 1} + \sqrt{x + 3} & = 3 \\ {(\sqrt{2 x - 1})}^{2} & = {(3 - \sqrt{x + 3})}^{2} \\ 2 x - 1 & = 9 - 6 \sqrt{x + 3} + x + 3 \\ 2 x - 1 - 12 - x & = - 6 \sqrt{x + 3} \\ {(x - 13)}^{2} & = {(- 6 \sqrt{x + 3})}^{2} \\ x^{2} - 26 x + 169 & = 36 (x + 3) \\ x^{2} - 26 x + 169 & = 36 x + 108 \\ x^{2} - 26 x + 169 - 36 x - 108 & = 0 \\ x^{2} - 62 x + 61 & = 0 \\ (x - 61) (x - 1) & = 0 \\ x - 1 & = 0 \\ x_{1} & = 1 \\ x - 61 & = 0 \\ x_{2} & = 61 \\ V e r i f i c a c i ó n : \\ S e a x = 1 \\ \sqrt{2 (1) - 1} + \sqrt{1 + 3} & = 3 \\ \sqrt{2 - 1} + \sqrt{4} & = 3 \\ 1 + 2 & = 3 \\ 3 & = 3 \Rightarrow x = 1 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = 61 \\ \sqrt{2 (61) - 1} + \sqrt{61 + 3} & = 3 \\ \sqrt{122 - 1} + \sqrt{64} & = 3 \\ \sqrt{121} + 8 & = 3 \\ 11 + 8 & = 3 \\ 18 & \neq 3 \Rightarrow x = 61 n o s a t i s f a c e l a e c u a c i ó n, p o r t a n t o e s u n a s o l u c i ó n e x t r a ñ a \end{matrix}$
$\begin{matrix} \sqrt{x - 3} + \sqrt{2 x + 1} - 2 \sqrt{x} & = 0 \\ {(\sqrt{x - 3} + \sqrt{2 x + 1})}^{2} & = {(2 \sqrt{x})}^{2} \\ x - 3 + 2 \sqrt{(x - 3) (2 x + 1)} + 2 x + 1 & = 4 x \\ 2 \sqrt{2 x^{2} + x - 6 x - 3} & = 4 x - 3 x + 2 \\ {(2 \sqrt{2 x^{2} - 5 x - 3})}^{2} & = {(x + 2)}^{2} \\ 4 (2 x^{2} - 5 x - 3) & = x^{2} + 4 x + 4 \\ 8 x^{2} - 20 x - 12 - x^{2} - 4 x - 4 & = 0 \\ 7 x^{2} - 24 x - 16 & = 0 \\ 7 x^{2} - 28 x + 4 x - 16 & = 0 \\ 7 x (x - 4) + 4 (x - 4) & = 0 \\ (7 x + 4) (x - 4) & = 0 \\ x - 4 & = 0 \\ x_{1} & = 4 \\ 7 x + 4 & = 0 \\ 7 x & = - 4 \\ x_{2} & = - \frac{4}{7} \\ V e r i f i c a c i ó n : \\ S e a x = 4 \\ \sqrt{4 - 3} + \sqrt{2 (4) + 1} - 2 \sqrt{4} & = 0 \\ \sqrt{1} + \sqrt{8 + 1} - 2 (2) & = 0 \\ 1 + 3 - 4 & = 0 \\ 0 & = 0 \Rightarrow x = 4 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = - \frac{4}{7} \\ \sqrt{- \frac{4}{7} - 3} + \sqrt{2 (- \frac{4}{7}) + 1} - 2 \sqrt{- \frac{4}{7}} & = 0 \\ \sqrt{\frac{- 4 - 21}{7}} + \sqrt{- \frac{8}{7} + 1} - 2 \sqrt{- \frac{4}{7}} & = 0 \\ \sqrt{\frac{- 25}{7}} + \sqrt{\frac{- 8 + 7}{7}} - 2 \frac{2}{\sqrt{7} i} & = 0 \\ \frac{5}{\sqrt{7} i} + \frac{1}{\sqrt{7} i} - 2 \frac{2}{\sqrt{7} i} & = 0 \\ \frac{5}{\sqrt{7} i} + \frac{1}{\sqrt{7} i} - \frac{4}{\sqrt{7} i} & \neq 0 \Rightarrow x = - \frac{4}{7} n o s a t i s f a c e l a e c u a c i ó n, p o r t a n t o e s u n a s o l u c i ó n e x t r a ñ a \end{matrix}$
$\begin{matrix} \sqrt{5 x - 1} - \sqrt{3 - x} & = \sqrt{2 x} \\ {(\sqrt{5 x - 1} - \sqrt{3 - x})}^{2} & = {(\sqrt{2 x})}^{2} \\ 5 x - 1 - 2 \sqrt{(5 x - 1) (3 - x)} + 3 - x & = 2 x \\ - 2 \sqrt{(5 x - 1) (3 - x)} & = 2 x - 4 x - 2 \\ - 2 \sqrt{15 x - 5 x^{2} - 3 + x} & = - 2 x - 2 \\ D i v i d i e n d o l a e c u a c i ó n p a r a - 2 \\ {(\sqrt{16 x - 5 x^{2} - 3})}^{2} & = {(x + 1)}^{2} \\ 16 x - 5 x^{2} - 3 & = x^{2} + 2 x + 1 \\ 16 x - 5 x^{2} - 3 - x^{2} - 2 x - 1 & = 0 \\ - 6 x^{2} + 14 x - 4 & = 0 \\ D i v i d i e n d o l a e c u a c i ó n p a r a - 2 \\ 3 x^{2} - 7 x + 2 & = 0 \\ 3 x^{2} - 6 x - x + 2 & = 0 \\ 3 x (x - 2) - (x - 2) & = 0 \\ (3 x - 1) (x - 2) & = 0 \\ x - 2 & = 0 \\ x_{1} & = 2 \\ 3 x - 1 & = 0 \\ 3 x & = 1 \\ x_{2} & = \frac{1}{3} \\ V e r i f i c a c i ó n : \\ S e a x = 2 \\ \sqrt{5 (2) - 1} - \sqrt{3 - 2} & = \sqrt{2 (2)} \\ \sqrt{10 - 1} - \sqrt{1} & = \sqrt{2^{2}} \\ \sqrt{9} - 1 & = 2 \\ 3 - 1 & = 2 \\ 2 & = 2 \Rightarrow x = 2 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = \frac{1}{3} \\ \sqrt{5 (\frac{1}{3}) - 1} - \sqrt{3 - \frac{1}{3}} & = \sqrt{2 (\frac{1}{3})} \\ \sqrt{\frac{5}{3} - 1} - \sqrt{\frac{9 - 1}{3}} & = \sqrt{\frac{2}{3}} \\ \sqrt{\frac{5 - 3}{3}} - \sqrt{\frac{8}{3}} & = \sqrt{\frac{2}{3}} \\ \sqrt{\frac{2}{3}} - \sqrt{\frac{8}{3}} & = \sqrt{\frac{2}{3}} \\ - \sqrt{\frac{8}{3}} & \neq 0 \Rightarrow x = \frac{1}{3} n o s a t i s f a c e l a e c u a c i ó n, p o r t a n t o e s u n a s o l u c i ó n e x t r a ñ a \end{matrix}$
$\begin{matrix} \sqrt{3 x + 1} + \sqrt{5 x} & = \sqrt{16 x + 1} \\ {(\sqrt{3 x + 1} + \sqrt{5 x})}^{2} & = {(\sqrt{16 x + 1})}^{2} \\ 3 x + 1 + 2 \sqrt{5 x (3 x + 1)} + 5 x & = 16 x + 1 \\ 2 \sqrt{15 x^{2} + 5 x} & = 16 x - 8 x \\ 2 \sqrt{15 x^{2} + 5 x} & = 8 x \\ D i v i d i e n d o l a e c u a c i ó n p a r a 2 \\ {(\sqrt{15 x^{2} + 5 x})}^{2} & = {(4 x)}^{2} \\ 15 x^{2} + 5 x & = 16 x^{2} \\ 0 & = 16 x^{2} - 15 x^{2} - 5 x \\ x^{2} - 5 x & = 0 \\ x (x - 5) & = 0 \\ x_{1} & = 0 \\ x - 5 & = 0 \\ x_{2} & = 5 \\ V e r i f i c a c i ó n : \\ S e a x = 0 \\ \sqrt{3 (0) + 1} + \sqrt{5 (0)} & = \sqrt{16 (0) + 1} \\ \sqrt{1} + 0 & = \sqrt{0 + 1} \\ 1 & = 1 \Rightarrow x = 0 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = 5 \\ \sqrt{3 (5) + 1} + \sqrt{5 (5)} & = \sqrt{16 (5) + 1} \\ \sqrt{15 + 1} + \sqrt{5^{2}} & = \sqrt{80 + 1} \\ \sqrt{16} + 5 & = \sqrt{81} \\ 4 + 5 & = 9 \\ 9 & = 9 \Rightarrow x = 5 s i s a t i s f a c e l a e c u a c i ó n \end{matrix}$
$\begin{matrix} \sqrt{2 x + \sqrt{4 x - 3}} & = 3 \\ {(\sqrt{2 x + \sqrt{4 x - 3}})}^{2} & = 3^{2} \\ 2 x + \sqrt{4 x - 3} & = 9 \\ {(\sqrt{4 x - 3})}^{2} & = {(9 - 2 x)}^{2} \\ 4 x - 3 & = 81 - 36 x + 4 x^{2} \\ 0 & = 81 - 36 x - 4 x + 3 + 4 x^{2} \\ 4 x^{2} - 40 x + 84 & = 0 \\ D i v i d i e n d o l a e c u a c i ó n p a r a 4 \\ x^{2} - 10 x + 21 & = 0 \\ (x - 7) (x - 3) & = 0 \\ x - 7 & = 0 \\ x_{1} & = 7 \\ x - 3 & = 0 \\ x_{2} & = 3 \\ V e r i f i c a c i ó n : \\ S e a x = 3 \\ \sqrt{2 (3) + \sqrt{4 (3) - 3}} & = 3 \\ \sqrt{6 + \sqrt{12 - 3}} & = 3 \\ \sqrt{6 + \sqrt{9}} & = 3 \\ \sqrt{6 + 3} & = 3 \\ \sqrt{9} & = 3 \\ 3 & = 3 \Rightarrow x = 3 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = 7 \\ \sqrt{2 (7) + \sqrt{4 (7) - 3}} & = 3 \\ \sqrt{14 + \sqrt{28 - 3}} & = 3 \\ \sqrt{14 + \sqrt{25}} & = 3 \\ \sqrt{14 + 5} & = 3 \\ \sqrt{19} & \neq 3 \Rightarrow x = 7 n o s a t i s f a c e l a e c u a c i ó n, p o r t a n t o e s u n a s o l u c i ó n e x t r a ñ a \end{matrix}$
$\begin{matrix} \sqrt{x + 3} + \frac{6}{\sqrt{x + 3}} & = 5 \\ \frac{{(\sqrt{x + 3})}^{2} + 6}{\sqrt{x + 3}} & = 5 \\ \frac{x + 3 + 6}{\sqrt{x + 3}} & = 5 \\ {(x + 9)}^{2} & = {(5 \sqrt{x + 3})}^{2} \\ x^{2} + 18 x + 81 & = 25 (x + 3) \\ x^{2} + 18 x + 81 & = 25 x + 75 \\ x^{2} + 18 x + 81 - 25 x - 75 & = 0 \\ x^{2} - 7 x + 6 & = 0 \\ (x - 6) (x - 1) & = 0 \\ x - 6 & = 0 \\ x_{1} & = 6 \\ x - 1 & = 0 \\ x_{2} & = 1 \\ V e r i f i c a c i ó n : \\ S e a x = 6 \\ \sqrt{6 + 3} + \frac{6}{\sqrt{6 + 3}} & = 5 \\ \sqrt{9} + \frac{6}{\sqrt{9}} & = 5 \\ 3 + \frac{}{3} & = 5 \\ 3 + 2 & = 5 \\ 5 & = 5 \Rightarrow x = 6 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = 1 \\ \sqrt{1 + 3} + \frac{6}{\sqrt{1 + 3}} & = 5 \\ \sqrt{4} + \frac{6}{\sqrt{4}} & = 5 \\ 2 + \frac{}{2} & = 5 \\ 2 + 3 & = 5 \\ 5 & = 5 \Rightarrow x = 1 s i s a t i s f a c e l a e c u a c i ó n d a d a \end{matrix}$
$\begin{matrix} \sqrt{x} + \frac{4}{\sqrt{x}} & = 5 \\ \frac{x + 4}{\sqrt{x}} & = 5 \\ {(x + 4)}^{2} & = {(5 \sqrt{x})}^{2} \\ x^{2} + 8 x + 16 & = 25 x \\ x^{2} + 8 x + 16 - 25 x & = 0 \\ x^{2} - 17 x + 16 & = 0 \\ (x - 16) (x - 1) & = 0 \\ x - 1 & = 0 \\ x_{1} & = 1 \\ x - 16 & = 0 \\ x_{2} & = 16 \\ V e r i f i c a c i ó n : \\ S e a x = 1 \\ \sqrt{1} + \frac{4}{\sqrt{1}} & = 5 \\ 1 + 4 & = 5 \\ 5 & = 5 \Rightarrow x = 1 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = 16 \\ \sqrt{16} + \frac{4}{\sqrt{16}} & = 5 \\ 4 + \frac{4}{4} & = 5 \\ 4 + 1 & = 5 \\ 5 & = 5 \Rightarrow x = 1 s i s a t i s f a c e l a e c u a c i ó n d a d a \end{matrix}$
$\begin{matrix} 2 \sqrt{x} & = \sqrt{x + 7} + \frac{8}{\sqrt{x + 7}} \\ 2 \sqrt{x} & = \frac{{(\sqrt{x + 7})}^{2} + 8}{\sqrt{x + 7}} \\ 2 \sqrt{x} & = \frac{x + 7 + 8}{\sqrt{x + 7}} \\ {(2 \sqrt{x (x + 7)})}^{2} & = {(x + 15)}^{2} \\ 4 (x^{2} + 7 x) & = x^{2} + 30 x + 225 \\ 4 x^{2} + 28 x & = x^{2} + 30 x + 225 \\ 4 x^{2} + 28 x - x^{2} - 30 x - 225 & = 0 \\ 3 x^{2} - 2 x - 225 & = 0 \\ 3 x^{2} - 27 x + 25 x - 225 & = 0 \\ 3 x (x - 9) + 25 (x - 9) & = 0 \\ (3 x + 25) (x - 9) & = 0 \\ x - 9 & = 0 \\ x_{1} & = 9 \\ 3 x + 25 & = 0 \\ 3 x & = - 25 \\ x_{2} & = - \frac{25}{3} \\ V e r i f i c a c i ó n : \\ S e a x = 9 \\ 2 \sqrt{9} & = \sqrt{9 + 7} + \frac{8}{\sqrt{9 + 7}} \\ 2 (3) & = \sqrt{16} + \frac{8}{\sqrt{16}} \\ 6 & = 4 + \frac{}{4} \\ 6 & = 4 + 2 \\ 6 & = 6 \Rightarrow x = 9 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = - \frac{25}{3} \\ 2 \sqrt{- \frac{25}{3}} & = \sqrt{- \frac{25}{3} + 7} + \frac{8}{\sqrt{- \frac{25}{3} + 7}} \\ 2 \frac{5}{\sqrt{3} i} & = \sqrt{\frac{- 25 + 21}{3}} + \frac{8}{\sqrt{\frac{- 25 + 21}{3}}} \\ 2 \frac{5}{\sqrt{3} i} & = \sqrt{\frac{- 4}{3}} + \frac{8}{\sqrt{\frac{- 4}{3}}} \\ 2 \frac{5}{\sqrt{3} i} & = \frac{2}{\sqrt{3} i} + \frac{8}{\frac{2}{\sqrt{3} i}} \\ 2 \frac{5}{\sqrt{3} i} & \neq \frac{2}{\sqrt{3} i} + \frac{8 \sqrt{3} i}{2} \Rightarrow x = - \frac{25}{3} n o s a t i s f a c e l a e c u a c i ó n, p o r t a n t o e s u n a s o l u c i ó n e x t r a ñ a \end{matrix}$
$\begin{matrix} \sqrt{x + \sqrt{x + 8}} & = 2 \sqrt{x} \\ {(\sqrt{x + \sqrt{x + 8}})}^{2} & = {(2 \sqrt{x})}^{2} \\ x + \sqrt{x + 8} & = 4 x \\ \sqrt{x + 8} & = 4 x - x \\ {(\sqrt{x + 8})}^{2} & = {(3 x)}^{2} \\ x + 8 & = 9 x^{2} \\ 9 x^{2} - x - 8 & = 0 \\ 9 x^{2} - 9 x + 8 x - 8 & = 0 \\ 9 x (x - 1) + 8 (x - 1) & = 0 \\ (9 x + 8) (x - 1) & = 0 \\ x - 1 & = 0 \\ x_{1} & = 1 \\ 9 x + 8 & = 0 \\ 9 x & = - 8 \\ x_{2} & = - \frac{8}{9} \\ V e r i f i c a c i ó n : \\ S e a x = 1 \\ \sqrt{1 + \sqrt{1 + 8}} & = 2 \sqrt{1} \\ \sqrt{1 + \sqrt{9}} & = 2 \\ \sqrt{1 + 3} & = 2 \\ \sqrt{4} & = 2 \\ 2 & = 2 \Rightarrow x = 1 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = - \frac{8}{9} \\ \sqrt{- \frac{8}{9} + \sqrt{- \frac{8}{9} + 8}} & = 2 \sqrt{- \frac{8}{9}} \\ \sqrt{- \frac{8}{9} + \sqrt{\frac{- 8 + 72}{9}}} & = 2 \frac{\sqrt{8}}{3} i \\ \sqrt{- \frac{8}{9} + \sqrt{\frac{64}{9}}} & = 2 \frac{\sqrt{8}}{3} i \\ \sqrt{- \frac{8}{9} + \frac{8}{3}} & = 2 \frac{\sqrt{8}}{3} i \\ \sqrt{\frac{- 8 + 24}{9}} & = 2 \frac{\sqrt{8}}{3} i \\ \sqrt{\frac{16}{9}} & = 2 \frac{\sqrt{8}}{3} i \\ \frac{4}{5} & \neq 2 \frac{\sqrt{8}}{3} i \Rightarrow x = - \frac{8}{9} n o s a t i s f a c e l a e c u a c i ó n, p o r t a n t o e s u n a s o l u c i ó n e x t r a ñ a \end{matrix}$
$\begin{matrix} \sqrt{6 - x} + \sqrt{x + 7} - \sqrt{12 x + 1} & = 0 \\ {(\sqrt{6 - x} + \sqrt{x + 7})}^{2} & = {(\sqrt{12 x + 1})}^{2} \\ 6 - x + 2 \sqrt{(6 - x) (x + 7)} + x + 7 & = 12 x + 1 \\ 2 \sqrt{6 x + 42 - x^{2} - 7 x} & = 12 x + 1 - 13 \\ 2 \sqrt{42 - x^{2} - x} & = 12 x - 12 \\ D i v i d i e n d o l a e c u a c i ó n p a r a 2 \\ {(\sqrt{42 - x^{2} - x})}^{2} & = {(6 x - 6)}^{2} \\ 42 - x^{2} - x & = 36 x^{2} - 72 x + 36 \\ 42 - x^{2} - x - 36 x^{2} + 72 x - 36 & = 0 \\ - 37 x^{2} + 71 x + 6 & = 0 \\ 37 x^{2} - 71 x - 6 & = 0 \\ 37 x^{2} - 74 x + 3 x - 6 & = 0 \\ 37 x (x - 2) + 3 (x - 2) & = 0 \\ (37 x + 3) (x - 2) & = 0 \\ x - 2 & = 0 \\ x_{1} & = 2 \\ 37 x + 3 & = 0 \\ 37 x & = - 3 \\ x_{2} & = - \frac{3}{37} \\ V e r i f i c a c i ó n : \\ S e a x = 2 \\ \sqrt{6 - 2} + \sqrt{2 + 7} - \sqrt{12 (2) + 1} & = 0 \\ \sqrt{4} + \sqrt{9} - \sqrt{24 + 1} & = 0 \\ 2 + 3 - \sqrt{25} & = 0 \\ 5 - 5 & = 0 \\ 0 & = 0 \Rightarrow x = 2 s i s a t i s f a c e l a e c u a c i ó n d a d a \\ S e a x = - \frac{3}{37} \\ \sqrt{6 - (- \frac{3}{37})} + \sqrt{- \frac{3}{37} + 7} - \sqrt{12 (- \frac{3}{37}) + 1} & = 0 \\ \sqrt{6 + \frac{3}{37}} + \sqrt{\frac{- 3 + 259}{37}} - \sqrt{- \frac{36}{37} + 1} & = 0 \\ \sqrt{\frac{222 + 3}{37}} + \sqrt{\frac{256}{37}} - \sqrt{\frac{- 36 + 37}{37}} & = 0 \\ \sqrt{\frac{225}{37}} + \frac{16 \sqrt{37}}{37} - \sqrt{\frac{1}{37}} & = 0 \\ \frac{15 \sqrt{37}}{37} + \frac{16 \sqrt{37}}{37} - \frac{\sqrt{37}}{37} & = 0 \\ \frac{\sqrt{37}}{37} (15 + 16 - 1) & = 0 \\ \frac{30 \sqrt{37}}{37} & \neq 0 \Rightarrow x = - \frac{3}{37} n o s a t i s f a c e l a e c u a c i ó n, p o r t a n t o e s u n a s o l u c i ó n e x t r a ñ a \end{matrix}$