To solve the equation $x^2 = 2^x$, we need to find all possible values of $x$ where the square of $x$ is equal to $2$ raised to the power of $x$.

Step 1: Check for simple solutions

Let's begin by testing some simple integer values for $x$:

• For $x = 0$: $x^2 = 0^2 = 0 \quad \text{and} \quad 2^x = 2^0 = 1$ So, $0^2 \neq 2^0$. Therefore, $x = 0$ is not a solution.

• For $x = 1$: $x^2 = 1^2 = 1 \quad \text{and} \quad 2^x = 2^1 = 2$ So, $1^2 \neq 2^1$. Therefore, $x = 1$ is not a solution.

• For $x = 2$: $x^2 = 2^2 = 4 \quad \text{and} \quad 2^x = 2^2 = 4$ So, $2^2 = 2^2$, meaning $x = 2$ is a solution.

• For $x = 4$: $x^2 = 4^2 = 16 \quad \text{and} \quad 2^x = 2^4 = 16$ So, $4^2 = 2^4$, meaning $x = 4$ is also a solution.

Step 2: Check for negative solutions

• For $x = -1$: $(-1)^2 = 1 \quad \text{and} \quad 2^{-1} = \frac{1}{2}$ So, $1 \neq \frac{1}{2}$. Therefore, $x = -1$ is not a solution.

• For $x = -2$: $(-2)^2 = 4 \quad \text{and} \quad 2^{-2} = \frac{1}{4}$ So, $4 \neq \frac{1}{4}$. Therefore, $x = -2$ is not a solution.

Step 3: Analyze the equation graphically or numerically

To explore more solutions beyond simple integer values, we can solve it graphically or numerically. The equation $x^2 = 2^x$ is a transcendental equation, which is generally difficult to solve algebraically. By plotting the two functions $y = x^2$ and $y = 2^x$, we can see their intersection points. From numerical analysis or a graph, we observe:

• $x = 2$
• $x = 4$

These two solutions are confirmed as the points where the curves intersect.

Conclusion:

The solutions to the equation $x^2 = 2^x$ are: $x = 2 \quad \text{and} \quad x = 4$

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Would you like a more detailed explanation of any steps? Here are five questions that could expand this discussion:

1. How can we graphically solve transcendental equations like this one?
2. Are there any methods for finding approximate solutions to transcendental equations?
3. Why are transcendental equations generally not solvable algebraically?
4. Can this equation have any complex number solutions?
5. How do numerical solvers like Newton's method work in finding solutions?

Tip: Transcendental equations often arise in mathematical modeling and physics, and numerical methods are a powerful way to approach them.