Question

1.)What number needs to be added to both sides of the equation in order to complete the square? x2+16x=18


answer is 64

x^2+16x+64=18+64

2.)Solve for x over the complex numbers.

x2+10x+41=0

answer is x=-5+4i and -5-4i

3.)What is the factored form of the expression over the complex numbers?

16x2+9y2
answer is (4x+3iy)(4x-3iy)

Answer 1

Answer:

all of your answers are correct

1.) 64

2.) x= -5+4i and x= -5-4i

3.) (4x+3iy)(4x-3iy)

Answer 2

Answer:

1.

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

$(\frac{16}{2})^{2} =(8)^{2}=64$

Basically, we need to add the number 64 both sides

$x^{2} +16x+64=18+64$

2.

The given equation is

$x^{2} +10x+41=0$

We need to apply the quadratic formula to solve this equation

$x_{1,2} =\frac{-b(+-)\sqrt{b^{2}-4ac } }{2a}$

Where $a=1$, $b=10$ and $c=41$. Replacing these values, we have

$x_{1,2} =\frac{-10(+-)\sqrt{10^{2}-4(1)(41) } }{2(1)}\\x_{1,2} =\frac{-10(+-)\sqrt{100-164 } }{2}=\frac{-10(+-)\sqrt{-64} }{2}$

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

$x_{1,2}=\frac{-10(+-)\sqrt{64}i }{2}=\frac{-10(+-)8i}{2}\\ x_{1,2}=-5(+-)4i$

Therefore, the complex solutions are

$x_{1}=-5+4i\\ x_{1}=-5-4i$

3.

The given expression is

$16x^{2} +9y^{2}$

To solve this expression, remember that $i=\sqrt{-1}$

First, we expresse both squares uniformly,

$16x^{2} +9y^{2}=(4x)^{2}+(3y)^{2}$

But, we know that $-(-1)=1$, so

$(4x)^{2}+(3y)^{2}=(4x)^{2}-(-1)(3y)^{2}$

Then,

$(4x)^{2}-(-1)(3y)^{2}=(4x)^{2}-(3y)^{2}i^{2}$, because $i^{2}=-1$

Therefore, the expression with complex numbers is

$(4x)^{2}-(3iy)^{2}\\\therefore (4x+3iy)(4x-3iy)$