equation with exponential functions 2

Question

Solve the following equation over the real numbers: $$ (3+ \sqrt{5})^x + (3- \sqrt{5})^x=7 * 2^x $$

Answer 1

Hint: Let $$a=\sqrt{3+\sqrt{5}}\;\;\;\;{\rm and }\;\;\;\;\;b=\sqrt{3-\sqrt{5}}$$

Then $ab = 2$ so we have $$ a^{2x}+b^{2x} = 7a^xb^x$$

Let $t= a^x/b^x$ then we have $$t^2-7t+1=0$$

can you do it now?

Answer 2

Hint: $$(3+\sqrt{5})(3-\sqrt{5})=4$$