Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 43549 by peter frank last updated on 11/Sep/18

$$solvex:{(5+2\sqrt{6})}^{x^2-3}+{(5-2\sqrt{6})}^{x^2-3}=10$$

Commented by Cheyboy last updated on 12/Sep/18

$$(5+2\sqrt{6})(5-2\sqrt{6})=1$$$${\left[(5+2\sqrt{6}){(5-2\sqrt{6})}^{}\right]}^{{\mathrm{x}}^2-3}=1^{{\mathrm{x}}^2-3}$$$${(5-2\sqrt{6})}^{{\mathrm{x}}^2-3}=\frac{1}{{(5+2\sqrt{6})}^{{\mathrm{x}}^2-3}}$$$${(5+2\sqrt{6})}^{{\mathrm{x}}^2-3}+\frac{1}{{(5+2\sqrt{6})}^{{\mathrm{x}}^2-3}}=10$$$$\mathrm{m}={(5+2\sqrt{6})}^{{\mathrm{x}}^2-3}$$$$\mathrm{m}+\frac{1}{\mathrm{m}}=10$$$${\mathrm{m}}^2-10\mathrm{m}+1=0$$$$\mathrm{m}=5\pm 2\sqrt{6}$$$${(5+2\sqrt{6})}^{{\mathrm{x}}^2-3}={(5+2\sqrt{6})}^1$$$${\mathrm{x}}^2-3=1$$$${\mathrm{x}}^2=4$$$$\mathrm{x}=\pm 2$$

Answered by behi83417@gmail.com last updated on 12/Sep/18

$${(5+2\sqrt{6})}^{x^2-3}=a\Rightarrow {(5-2\sqrt{6})}^{x^2-3}=a^{-1}$$$$\Rightarrow a+a^{-1}=10\Rightarrow a^2-10a+1=0$$$$a=\frac{10\pm \sqrt{96}}{2}=5\pm 2\sqrt{6}$$$$\Rightarrow x^2-3=\pm 1\Rightarrow x^2=4,2\Rightarrow x=\pm 2,\pm \sqrt{2}.◼$$

Commented by peter frank last updated on 12/Sep/18

$$thankyou$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com