- #1
jya
- 2
- 0
I know we can solve e^x=x by the Lambert W function, but is it possible to solve the following equation:
a*(e^(2x)-e^x)+b*x=c
in terms of a, b, and c.
a*(e^(2x)-e^x)+b*x=c
in terms of a, b, and c.
Solving for x given a,b,c you mean? That would be the intersection of a line with a quadratic in e^x... that is $$ e^{x}\left ( e^x - 1\right ) = mx+k$$ ...where ##m=-b/a## and ##k=c/a##.solve: a*(e^(2x)-e^x)+b*x=c ... in terms of a, b, and c.
Simon Bridge said:Welcome to PF;
Solving for x given a,b,c you mean? That would be the intersection of a line with a quadratic in e^x... that is $$ e^{x}\left ( e^x - 1\right ) = mx+k$$ ...where ##m=-b/a## and ##k=c/a##.
And you want to find x given m and k.
That help?
The method you use to solve an equation depends on the type of equation you are working with. For linear equations, you can use the elimination or substitution method. For quadratic equations, you can use factoring, the quadratic formula, or completing the square. It is important to understand the properties and rules of each method in order to determine the most efficient way to solve the equation.
While a calculator can be a helpful tool in solving equations, it is important to first understand the steps and concepts behind solving the equation. Relying solely on a calculator can hinder your ability to fully understand and solve the equation on your own. It is recommended to only use a calculator as a tool to check your work.
To check if your solution is correct, you can substitute the solution back into the original equation and solve to see if the equation holds true. For example, if your solution is x = 3, you can plug in 3 for x in the original equation and see if both sides still equal each other. If they do, then your solution is correct.
If you get a negative or imaginary number as your solution, it means that the equation does not have a real number solution. This is common in quadratic equations, where the solutions may be complex numbers. In these cases, you can leave your answer in terms of the imaginary number or simplify it to a decimal approximation.
The best way to improve your equation-solving skills is through practice and understanding the underlying concepts. Make sure to review the properties and rules of each method and attempt a variety of equations to become familiar with the different types and how to approach them. Asking for help from a teacher or tutor can also be beneficial in understanding and improving your skills.