1.1 Lösning 9
\(\begin{align} 2 & = - { x \over \sqrt{1-x^2} } & & | \;\; \cdot \sqrt{1-x^2} \\ 2 \cdot \sqrt{1-x^2} & = - \; x & & | \; (\;\;\;)^2 \\ 4 \cdot (1 - x^2) & = x^2 \\ 4 - 4\,x^2 & = x^2 & & | \; + 4\,x^2 \\ 4 & = 5\,x^2 & & | \; / \; 5 \\ {4 \over 5} & = x^2 & & | \; \sqrt{\;\;} \\ x_{1,2} & = \pm {2 \over \sqrt{5}} \\ x_1 & = {2 \over \sqrt{5}} \\ x_2 & = -{2 \over \sqrt{5}} \\ \end{align}\)
Prövning:
Först prövar vi \( x_1 = {2 \over \sqrt{5}} \)
VL\[ 2\, \]
HL\[ - { {2 \over \sqrt{5}} \over \sqrt{1-{4 \over 5}} } = - { {2 \over \sqrt{5}} \over \sqrt{1 \over 5} } = - { {2 \over \sqrt{5}} \over {1 \over \sqrt{5}} } = - { {2 \over \sqrt{5}} \over {\sqrt{1} \over \sqrt{5}} } = - { {2 \cdot \sqrt{5}} \over {\sqrt{1} \cdot \sqrt{5}} } = - { 2 \over \sqrt{1} } = -\,2 \]
VL \( \not= \) HL \( \Rightarrow\; x_1 = {2 \over \sqrt{5}} \) är en falsk rot.
Sedan prövar vi roten \( x_2 = - {2 \over \sqrt{5}} \):
VL\[ 2\, \]
HL\[ - { {-2 \over \sqrt{5}} \over \sqrt{1-{4 \over 5}} } = { {2 \over \sqrt{5}} \over \sqrt{1 \over 5} } = { {2 \over \sqrt{5}} \over {1 \over \sqrt{5}} } = { {2 \over \sqrt{5}} \over {\sqrt{1} \over \sqrt{5}} } = { {2 \cdot \sqrt{5}} \over {\sqrt{1} \cdot \sqrt{5}} } = { 2 \over \sqrt{1} } = 2 \]
VL = HL \( \Rightarrow\; x_2 = - {2 \over \sqrt{5}} \) är en sann rot.
Svar: Ekvationen
\[ 2 = - { x \over \sqrt{1-x^2} } \]
har den enda lösningen
- \[ x = - {2 \over \sqrt{5}} \]
