Why and when is i used in algebra

Question

Select all roots of the equation given by x²-4x+5=0

1. 2-i
2. 2+i
3. 1+2i
4. 1-2i

check all that apply

Answer 1

±i is the square root of -1.

We need to complete the square to find the roots of the equation:

x² - 4x + 5 = 0.

We start by subtracting 5 from both sides:

x² - 4x = -5.

Now we have to create a trinomial square on the left side of the equation by finding a value that is equal to the square of half of b, the coefficient of x (in this case b = -4):

(b/2)² = (-4/2)² = (-2)² = 4,

and add that term to both sides:

x² - 4x + 4 = -5 + 4,

and simplify:

x² - 4x + 4 = -1.

Now we can factor the left side:

(x - 2)² = -1,

and solve for x by taking the square root of both sides:

x - 2 = √(-1),

and isolating x by adding 2 to both sides:

x = √(-1) + 2.

Now, since we know that the square root of -1 is ±i , we get:

x = 2 ± i.

Therefore, answer options 1. and 2. are correct.

Answer 2

Greetings! Lets solve this shall we ?

We must find the roots of the polynomial x²-4x+5=0. We notice this polynomial is not factorable algebraically because we cannot find factors of 5 that will add together to obtain -4. We use the quadratic formula instead where a = 1, b = -4 and c = 5 such that

-b ± √ b² - 4ac -(-4) ± √(-4)² - 4(1)(5) 4 ± √(-4)

---------------------- = ------------------------------ = ------------- = (2 + i), (2 - i)

2a 2(1) 2

After simplifying this, it becomes (2+i) and (2-i). Why ? The square root of (-4) = ± 2 i because the square root of -1 is i by definition.

I hope this helped!

Answer 3

The roots are 2±i.