Solving Transcendental Equation
The image shows a mathematical equation to solve:
\[ e^{4x} - 7x + 11 = 20 \]
We want to solve for \( x \). To start, we can simplify the equation by moving all terms to one side to set the equation to zero.
First, subtract 20 from both sides of the equation:
\[ e^{4x} - 7x + 11 - 20 = 0 \]
Now simplify:
\[ e^{4x} - 7x - 9 = 0 \]
This is a transcendental equation because it involves an algebraic part \((-7x - 9)\) and a transcendental part \((e^{4x})\). These equations are not typically solvable using elementary algebraic methods and often require numerical methods to find an approximate solution.
Common numerical methods for solving transcendental equations include the Newton-Raphson method, the bisection method, or using software or graphing calculators.
If an exact solution is not required, you could find approximate solutions using a graphing calculator or software by plotting the function \[ f(x) = e^{4x} - 7x - 9 \] and looking for the \( x \)-values where \( f(x) \) crosses the \( x \)-axis. These are the real roots of the equation.
To solve this accurately, you would generally use a calculator or a computational tool that can handle transcendental equations.
