The quadratic equation $${x^2} - 4x + 4 = 0$$ is to be solved numerically, starting with the initial guess $${x_0} = 3.$$ The Newton- Raphson method is applied once to get a new estimate and then the Secant method is applied once using in the initial guess and this new estimate. The estimated value of the root after the application of the Secant method is ________.

In Newton-Raphson iterative method, the initial guess value $$\left( {{x_{ini}}} \right)$$ is considered as zero while finding the roots of the equation: $$\,f\left( x \right) = - 2 + 6x - 4{x^2} + 0.5{x^3}.\,\,\,$$ The correction, $$\Delta x,$$ to be added to $${{x_{ini}}}$$ in the first iteration is __________.

For step-size, $$\Delta x = 0.4,$$ the value of following integral using Simpson's $$1/3$$ rule is ______

There is no value of $$x$$ that can simultaneously satisfy both the given equations. Therefore, find the 'least squares error' solution to the two equations, i.e., find the value of $$x$$ that minimizes the sum of squares of the errors in the two equations
$$2x=3$$
$$4x=1$$