Final answer:
Using the Newton-Raphson method, we approximate the root of a quadratic equation by iterating the formula x₁ = x₀ - f(x₀)/f'(x₀) starting from the initial guess x₀=3 and continuing until the tolerance of 10⁻⁶ is met.
Step-by-step explanation:
The Newton-Raphson method is an iterative numerical method used to find the roots of a real-valued function. The student's question involves computing the root of a specific quadratic equation, which typically can be solved using the quadratic formula. However, here we are asked to use the Newton-Raphson method with an initial guess and a specified tolerance. The given quadratic equation is x² + 1.2 x 10⁻²x - 6.0 × 10⁻³ = 0. The student has not provided the exact function to use for Newton-Raphson, which might be either f(x) = x² + 1.2 x 10⁻²x - 6.0 × 10⁻³ or another related function.
However, using the provided example and Newton-Raphson iteration formula, the process to follow generally involves calculating the derivative of the function, then applying the iteration x₁ = x₀ - f(x₀)/f'(x₀), and repeating this process until the absolute value of the function at the new approximate root is less than the tolerance.
For example, if the function was indeed f(x) = x² + 1.2 x 10⁻²x - 6.0 × 10⁻³, we would calculate the derivative f'(x) = 2x + 1.2 x 10⁻² and then apply the iterative formula starting with x₀ = 3 to find an approximate root that satisfies the tolerance condition.