To solve the equation $2^x + x = 4$, we can use numerical methods as it does not have a straightforward algebraic solution. Here’s a step-by-step approach:

1. Rewriting the Equation: Consider the function $f(x) = 2^x + x - 4$.

2. Finding the Root Numerically:

   • Evaluate $f(x)$ at different values of $x$ to find an approximate solution.
   • Use methods like the bisection method or Newton's method to refine the solution.

Let's use a numerical approach to find the solution:

1. Initial Estimates:

   • $f(0) = 2^0 + 0 - 4 = -3$
   • $f(1) = 2^1 + 1 - 4 = -1$
   • $f(2) = 2^2 + 2 - 4 = 2$

   Since $f(1)$ and $f(2)$ have opposite signs, a root lies between $x = 1$ and $x = 2$.

2. Refinement: Using the bisection method:

   • Midpoint $x = 1.5$, $f(1.5) = 2^{1.5} + 1.5 - 4 \approx -0.17$.
   • Midpoint $x = 1.75$, $f(1.75) = 2^{1.75} + 1.75 - 4 \approx 0.59$.
   • Since $f(1.5)$ and $f(1.75)$ have opposite signs, a root lies between $x = 1.5$ and $x = 1.75$.

Continuing this process will eventually lead us to the root. Using numerical solvers, we find the solution is approximately:

$x \approx 1.386$

Would you like a more detailed explanation or steps on solving it numerically?

Questions for You to Ask

1. How do numerical methods like bisection and Newton's method work?
2. What are the common numerical methods for solving transcendental equations?
3. Can you explain the concept of a function root?
4. How do you determine initial estimates for numerical methods?
5. What are the advantages and limitations of numerical methods?

Tip

For transcendental equations, graphical methods can help you visualize where the function crosses the x-axis, providing a good starting point for numerical solutions.