# Find square root

#### aisha

does anyone know how to find the square root of 5+12i ?

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#### Hurkyl

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Try asking yourself the same question written differently:

Can you find a number z such that z*z = 5 + 12i?

#### aisha

Hurkyl said:
Try asking yourself the same question written differently:

Can you find a number z such that z*z = 5 + 12i?
That sounds easier but im not sure how to figure that out.

This was a multiple choice question so the answer is 2+i
I dont know how they got that!!

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#### phreak

aisha said:
This was a multiple choice question so the answer is 2+i
Actually... no, that's wrong. The answer is 3 + 2i.

#### Hurkyl

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If it's multiple choice, then it's even easier.

You say one of the choices was 2+i. Well, what is (2+i)*(2+i)? Is it 5+12i?

Do you know of a way to write down an arbitrary complex number, z, in terms of two (real) variables? If so, then you can compute what z^2 is...

#### aisha

phreak said:
Actually... no, that's wrong. The answer is 3 + 2i.
can u plz tell me how you got that answer?

#### xLaser

remember i^2 = -1 .

#### nolachrymose

Well, you know that z is a complex number in the form a+bi. Therefore, you can say this:

$$(a+bi)^2=5+12i$$
$$a^2+2abi-b^2=5+12i$$

Now, since a and b are real, you know a^2-b^2 = 5, and 2abi = 12i. Therefore, you can solve the system of equations to find a and b, thus finding z.

#### phreak

aisha said:
can u plz tell me how you got that answer?
Unfortunately, it was only a matter of guessing (and reverse factoring).

I simply used: (ai + b)^2 = 12i + 5

Expanding the equation: (ai)^2 + (ab)i + b^2 = 12i + 5

Knowing that i^2 = -1, this equation can be simplified further:

-(a^2) + 2(ab)i + b^2 = 12i + 5

so... therefore:

a*b must equal 12 and 2(b^2 - a^2) must equal 5

Make a system of equations and solve:

2ab = 12
b^2 - a^2 = 5

Solving, we find that a = 2 and b = 3

Input them into the original expanded equation (-(a^2) + 2(ab)i + b^2 = 12i + 5)

So:

-(2^2) + 2(2*3)i + 3^2 = 12i + 5

-4 + 12i + 9

12i + 5

There's an easier method to this, probably... sorry I can't really help you out.

(EDIT: This came 3 min. after nolachrymose made a post... I'm really slow at this.)

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#### aisha

Phreak everything makes sense except for where did u get a=2 b=3? what did u solve and how? Im so dumb sorry lol I really dont like math.

Thanks Nolachrymose I would have never figured this out without u and Phreak thanks guys, Last edited:

#### Gokul43201

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Science Advisor
Gold Member
You have ab = 6, so b = 6/a
And b^2 - a^2 = 5, so 36/a^2 - a^2 = 5. This is just a quadratic in a^2. Solve it to find a^2, and hence a and b.

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