The Quadratic in Disguise

Solve the equation you found in Phase 1.

Use your solution(s) to find the corresponding value(s) of \(x\).

Substitute each value of \(x\) back into the original equation \(4^x-5(2^x)+4=0\) to confirm it works.

Use the method your group developed to solve \(9^x - 4(3^x) - 45 = 0\).

Before you finalize your answer(s) for \(x\): look carefully at every value you found for your substituted variable. Is each one actually possible?

Use your method on \(2^x + 2^{-x} = 1\). What equation do you end up with after substituting?

What you have doesn't look like the quadratics you've solved so far. Can your group find a way to turn it into one?

Solve the quadratic equation you found. What happens?

Given what you found in Task 4c, what can you conclude about the original equation \(2^x+2^{-x}=1\)? Is there any real value of \(x\) that makes it true?