Question

What is the product of -4 - 7i and its conjugate?

Answer 1

the conjugate of a+bi is a-bi

Answer 2

so the conjugate of -4-7i is -4-(-7i) or -4+7i

Answer 3

so, -33? ((Thank you so very much! Sorry, I missed something so simple!))

Answer 4

The product of a complex number and its conjugate is done so frequently that it is worth learning a little trick:

\[(a+bi)(a-bI) = a^2 + abi - abi b^2i^2 \]\[\qquad= a^2 - b^2i^2\]However, \(i=\sqrt{=-1} \implies i^2=-1\) so if we substitute \(-1\) for \(i^2\) we get
\[(a+bi)(a-bi) = a^2-b^2i^2 = a^2-b^2(-1) = a^2+b^2\]

Answer 5

You have \(a = -4\) and \(b = -7\)
What is \(a^2+b^2=\)

Answer 6

I completely fumble-fingered that first equation:

\[(a+bi)(a-bi) = a^2 + abi - abi -b^2i^2\]\[\qquad=a^2-b^2i^2\]

Answer 7

@michaelaford13 do you have a new, improved answer?