Question

The quadratic equation $ 2x^2 - 3x + 3 = 0 $ is to be solved numerically starting with initial guess $ x_0 = 2 $. The new estimate of $ x $ after the first iteration using the Newton-Raphson method is

• A. 1
• B. 2
• C. \( \frac{1}{152} \)
• D.

Hint:

The Newton-Raphson method is an iterative technique for finding successively better approximations to the roots of a real-valued function. If the initial guess is close enough to the true root, the method converges rapidly.

Answer 1

The Correct Option is A

The Newton-Raphson method is used to find the roots of a function. The iterative formula for the Newton-Raphson method is: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] Given the equation \( f(x) = 2x^2 - 3x + 3 \), we need to first compute \( f(x) \) and its derivative \( f'(x) \): \[ f(x) = 2x^2 - 3x + 3 \] \[ f'(x) = 4x - 3 \] Now, substitute the initial guess \( x_0 = 2 \) into the formula: \[ f(2) = 2(2)^2 - 3(2) + 3 = 8 - 6 + 3 = 5 \] \[ f'(2) = 4(2) - 3 = 8 - 3 = 5 \] Now, using the Newton-Raphson formula: \[ x_1 = 2 - \frac{5}{5} = 2 - 1 = 1 \]
Thus, the new estimate of \( x \) after the first iteration is 1.
Therefore, the correct answer is 1. 1.