Spud has a problem. He knows that the roots of a quadratic function are $x = 3 + 4i$ and $x = 3 - 4i$, but in order to get credit for the problem he was supposed to have written down the original equation. Unfortunately, he lost the paper with the original equation on it. Luckily, his friends are full of advice.

1. Alexia says, “Just remember when we made factors from the solutions. If you wanted $7$ and $4$ to be the answers, you used $y = \left(x-7\right)\left(x -4\right)$.” Use Alexia’s idea to write the equation of a quadratic function with which Spud could start.

   Step (1a):

   $y = \left(x − \left(3 + 4i\right)\right)\left(x − \left(3 − 4i\right)\right)$

   Step (2a):

   Step (3a):

   Step (4a):

   Answer (a):

   $y = x^{2} − 6x + 25$

   	$x$	$-\left(3+4i\right)$
   $x$
   $-\left(3-4i\right)$

   	$x$	$-\left(3+4i\right)$
   $x$	$x^2$	$-x\left(3+4i\right)$
   $-\left(3-4i\right)$	$-x\left(3-4i\right)$	$\left(3-4i\right)\left(3+4i\right)$

   	$x$	$-\left(3+4i\right)$
   $x$	$x^2$	$-3x-4xi$
   $-\left(3-4i\right)$	$-3x+4xi$	$25$

2. Hugo says, “No, no, no. You can do it that way, but that’s too complicated. I think you just start with $x = 3 + 4i$ and work backwards. So $x – 3 = 4i$, then, hmmm… Yeah, that’ll work.” Try Hugo’s method.

   Step (b):

   $\left(x − 3\right)^{2} = \left(4i\right)^{2}$

3. Whose method do you think Spud should use? Explain your choice.