Question

Помогите решить уравнение.

Answer 1

$sqrt{7 - x^{2}} + sqrt{6 + x^{2}} = 5\sqrt{7 - x^{2}} = 5 - sqrt{6 + x^{2}}\(sqrt{7 - x^{2}})^{2} = (5 - sqrt{6 + x^{2}})^{2}\7 - x^{2} = 25 - 10sqrt{6 + x^{2}} + 6 + x^{2}\2x^{2} + 24 = 10sqrt{6 + x^{2}} |:2\x^{2} + 12 = 5sqrt{6 + x^{2}}\(x^{2} + 12)^{2} =(5sqrt{6 + x^{2}})^{2}\x^{4} + 24x^{2} + 144 = 150 +25x^{2}\x^{4} - x^{2} - 6 = 0\left{begin{matrix}x_{1}^{2} + x^{2}_{2} = 1 & & \ x_{1}^{2} cdot x^{2}_{2} = -6 & &end{matrix}right.$

$left[begin{array}{ccc}x_{1}^{2} = 3 \ \x_{2}^{2} =-2\end{array}right\\left[begin{array}{ccc}x_{1} = pm sqrt{3} \ \x_{2} = oslash  \end{array}right$

Проверка:

$1) x = -sqrt{3}: sqrt{7 - (-sqrt{3})^{2}} + sqrt{6 + (-sqrt{3})^{2}} = sqrt{7 - 3} + sqrt{6 + 3} = sqrt{4} + sqrt{9} = \= 2 + 3 = 5;\\2) x = sqrt{3} : sqrt{7 - (sqrt{3})^{2}} + sqrt{6 + (sqrt{3})^{2}} = sqrt{7 - 3} + sqrt{6 + 3} = sqrt{4} + sqrt{9} = \= 2 + 3 = 5.$

Ответ: $pm sqrt{3}$