Let's analyze and solve the two given equations from the image:

1. Problem 2:

$2 \cdot 4^x - 5 \cdot 2^x + 2 - 6 = 0$

2. Problem 3:

$9^x - 2 \cdot 3^{x+1} - 27 = 0$

I'll start by solving the first one and then the second.

Solution to Problem 2:

We can rewrite $4^x$ as $(2^2)^x = 2^{2x}$, and solve the equation accordingly.

$2 \cdot 2^{2x} - 5 \cdot 2^x + 2 - 6 = 0$ Simplifying the constant terms: $2 \cdot 2^{2x} - 5 \cdot 2^x - 4 = 0$ Let $y = 2^x$, so the equation becomes: $2y^2 - 5y - 4 = 0$ Now, we can solve this quadratic equation using the quadratic formula.

Solution to Problem 3:

The second equation can be simplified using the fact that $9^x = (3^2)^x = 3^{2x}$.

$3^{2x} - 2 \cdot 3^{x+1} - 27 = 0$ Let $z = 3^x$, so the equation becomes: $z^2 - 2 \cdot 3z - 27 = 0$ This is a quadratic equation in terms of $z$, and we can solve it using the quadratic formula.

Let me calculate both quadratic equations and find the solutions.### Solutions:

1. For Problem 2:

The quadratic equation $2y^2 - 5y - 4 = 0$ has the following solutions: $y_1 = \frac{5 - \sqrt{57}}{4}, \quad y_2 = \frac{5 + \sqrt{57}}{4}$ Since $y = 2^x$, we'll need to check which of these solutions can be used to find a real value of $x$, as $2^x$ must be positive.

2. For Problem 3:

The quadratic equation $z^2 - 6z - 27 = 0$ gives: $z_1 = -3, \quad z_2 = 9$ Since $z = 3^x$, we can only consider the positive solution $z = 9$, as $3^x$ must also be positive. Therefore: $3^x = 9 \quad \Rightarrow \quad x = 2$

Would you like me to proceed with simplifying the solutions for $x$ in Problem 2, or do you have any other questions?

Here are some related questions for deeper understanding:

1. How does changing the base of exponential terms affect the solutions?
2. Why do we reject negative values for $2^x$ and $3^x$?
3. What methods can be used to solve quadratic equations besides the quadratic formula?
4. How can logarithms be applied to find $x$ after solving for $y$?
5. Can these types of exponential equations always be simplified using substitution?

Tip: When solving equations involving powers, always check if the base allows for negative values before discarding solutions.