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### Finding the Square Root of a Quadratic Function

```Date: 10/30/2002 at 15:09:12
From: Ryan MacRae
Subject: Finding the square root of a Quadratic Function.

I was receintly assinged the problem

Find the square root of 3+4i

I am in advanced college math and am totally stumped by this question.

Ryan
```

```
Date: 10/30/2002 at 17:46:55
From: Doctor Paul
Subject: Re: Finding the square root of a Quadratic Function.

If sqrt(3+4*i) = a + b*i where a and b are real numbers, then squaring
both sides gives:

(a + b*i) * (a + b*i) = 3 + 4*i

a^2 + 2*a*b*i - b^2 = 3 + 4*i

equating coefficients gives:

a^2 - b^2 = 3
and
2*a*b = 4

Solve for b in the second equation and substitute into the first
equation:

b = 2/a

so

a^2 - (2/a)^2 = 3

a^2 - 4/a^2 = 3

multiply both sides by a^2:

a^4 - 3*a^2 - 4 = 0

This is quadratic in a^2 so we make a substitution:

Now let x = a^2

So we have:

x^2 - 3*x - 4 = 0

(x-4)*(x+1) = 0

So x = 4 or x = -1

This gives

a = 2, -2, i, -i

We said above that a had to be real, so it must be the case that
a = 2 or a = -2.

Now, if a = 2 then b = 1

and  if a = -2 then b = -1

So we have:

sqrt(3 + 4*i) = (2 + i) or -(2 + i)

I hope this helps.  Please write back if you'd like to talk about
this some more.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
High School Imaginary/Complex Numbers

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