Question

Помогите, решить показательные уравнение!

Answer 1

ОДЗ :

x² - 2 ≥ 0

x ∈ (- ∞ ; - √2] ∪ [√2 ; + ∞)

$4^{\sqrt{x^{2}-2} +x}-5*2^{\sqrt{x^{2}-2 }+x-1 }=6\\\\4^{\sqrt{x^{2}-2 }+x } -\frac{5}{2}*2^{\sqrt{x^{2}-2 }+x }-6=0\\\\2*4^{\sqrt{x^{2}-2 }+x }-5*2^{\sqrt{x^{2}-2 }+x }-12=0\\\\2^{\sqrt{x^{2}-2 } +x}=t,t>0\\\\2t^{2}-5t-12=0\\\\D=(-5)^{2}-4*2*(-12)=25+96=121=11^{2}\\\\t_{1}=\frac{5+11}{4}=4\\\\t_{2}=\frac{5-11}{4}=-\frac{3}{2}<0$

$2^{\sqrt{x^{2}-2 } +x}=4\\\\2^{\sqrt{x^{2} -2}+x }=2^{2}\\\\\sqrt{x^{2}-2 }+x=2\\\\\sqrt{x^{2}-2 }=2-x\\\\(\sqrt{x^{2}-2 } )^{2}=(2-x)^{2} \\\\x^{2} -2=4-4x+x^{2}\\\\4x=6\\\\x=1,5$

Ответ : 1,5