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Use Newton's method to find all roots of the equation correct to six decimal places. e^x = 6 - 2x

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the corrected root for the equation e^x = 6 - 2x using Newton's method with an initial guess of x_0 = 1.0 is approximately 1.2517579.

Use Newton's method to find all roots of the equation correct to six decimal places: e^x = 6 - 2x.

Define the function: The equation is f(x) = e^x - (6 - 2x).

Derive the expression for the derivative f'(x): f'(x) = e^x + 2.

Set up the iteration formula for Newton's method: x_{n+1} = x_n - f(x_n)/f'(x_n).

Perform iterations: Starting with an initial guess, use the iteration formula until convergence.

Iterate until the difference between consecutive values is sufficiently small. For example, starting with an initial guess of x_0 = 1.0, x_1 = x_0 - f(x_0)/f'(x_0), x_2 = x_1 - f(x_1)/f'(x_1), and so on.

Now, use this converged value as a new initial guess and iterate again. The process is repeated until the value stabilizes. After several iterations, the root is found to be 1.2517579.

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