1.)what number needs to be added to both sides of the equation in order to complete the square? x2+16x=18answer is 64x^2+16x+64=18+642.)solve for x over the complex numbers. x2+10x+41=0answer is x=-5+4i and -5-4i3.)what is the factored form of the expression over the complex numbers? 16x2+9y2answer is (4x+3iy)(4x-3iy)

When we are completing squares, we need to divide by 2 the linear term and then find its square power, that's the term we need to add on both sides of the equality, as follows

Basically, we need to add the number 64 both sides

The given equation is

We need to apply the quadratic formula to solve this equation

Where , and . Replacing these values, we have

There we need to use complex number, to transform the subradical number in a positive number

Therefore, the complex solutions are

The given expression is

To solve this expression, remember that

First, we expresse both squares uniformly,

But, we know that , so

Then,

, because

Therefore, the expression with complex numbers is

step-by-step explanation:

f(x)=2x^2 -1 

translation is a function transformation that transform the function in a way of translating. by translating the function its graph is moved  up an down. moving the function f(x) 2 units to the left will be the new function g(x) with the equation: g(x)=f(x+2)=2(x+2)^2 -1